Upwind schemes for the wave equation in second-order form

Transcription

1 Upwnd schemes for the wave equaton n second-order form Jeffrey W. Banks a,,, Wllam D. Henshaw a, a Center for Appled Scentfc Computng, Lawrence Lvermore Natonal Laboratory, Lvermore, CA 9455, USA Abstract We develop new hgh-order accurate upwnd schemes for the wave equaton n second-order form. These schemes are developed drectly for the equatons n second-order form, as opposed to transformng the equatons to a frst-order hyperbolc system. The schemes are based on the soluton to a local Remann-type problem that uses d Alembert s eact soluton. We construct conservatve fnte dfference appromatons, although fnte volume appromatons are also possble. Hgh-order accuracy s obtaned usng a space-tme procedure whch requres only two dscrete tme levels. The advantages of our approach nclude effcency n both memory and speed together wth accuracy and robustness. The stablty and accuracy of the appromatons n one and two space dmensons are studed through normal-mode analyss. The form of the dsspaton and dsperson ntroduced by the schemes s elucdated from the modfed equatons. Upwnd schemes are mplemented and verfed n one dmenson for appromatons up to sth-order accuracy, and n two dmensons for appromatons up to fourth-order accuracy. Numercal computatons demonstrate the attractve propertes of the approach for solutons wth varyng degrees of smoothness. Keywords: second-order wave equatons, upwnd dscretzaton, Godunov methods, hgh-order accurate, fnte-dfference, fnte-volume Contents Introducton Constructon of an upwnd scheme for the second-order wave equaton n one dmenson 3 3 A general constructon for hgh-order upwnd schemes the one-dmensonal case 8 3. The upwnd flu for the second-order system A frst-order accurate upwnd scheme A second-order accurate scheme and a hgh-resoluton scheme A fourth-order accurate upwnd scheme A sth-order accurate upwnd scheme Upwnd schemes n two space dmensons 4. Dscretzaton n two space dmensons A frst-order accurate scheme n two dmensons A second-order accurate scheme and a hgh-resoluton scheme n two dmensons A fourth-order accurate scheme n two dmensons Accuracy, stablty, ponts per wavelength and remarks 7 Correspondng author. Malng address: Center for Appled Scentfc Computng, L-4, Lawrence Lvermore Natonal Laboratory, Lvermore, CA 9455, USA. Phone: Fa: Emal addresses: banks@llnl.gov (Jeffrey W. Banks), henshaw@llnl.gov (Wllam D. Henshaw) Ths work was performed under the auspces of the U.S. Department of Energy (DOE) by Lawrence Lvermore Natonal Laboratory under Contract DE-AC5-7NA7344 and by DOE contracts from the ASCR Appled Math Program. Preprnt submtted to Elsever September 6,

2 6 Numercal eamples 3 6. One-dmensonal travelng sne wave One-dmensonal top-hat problem A two-dmensonal surface wave problem A top-hat problem n two space dmensons Conclusons 37 Append A Hgh-order accurate stencls 38 Append A. Fourth-order accurate scheme Append A. Sth-order accurate scheme Introducton Upwnd methods for frst-order systems of hyperbolc of partal dfferental equatons (PDEs) have made a sgnfcant mpact on computatonal scence. They have facltated the smulaton of a wde varety of physcal problems (e.g. flud dynamcs, sold mechancs, electromagnetcs, plasma physcs, as well as other wave propagaton phenomena), usng a wde varety of numercal methods (e.g. fnte volume, fnte dfference, fnte element, spectral element and dscontnuous Galerkn). Upwnd methods generally ncorporate some aspects of the characterstc wave-structure of the hyperbolc system and thus nclude some of the attractve features of a method based on characterstcs. An alternatve approach to upwndng s based on the use of centered appromatons and the eplct addton of artfcal dsspaton (artfcal vscosty) or flters. Ths approach usually requres some tunng of the coeffcents of the artfcal dsspaton and for dffcult problems may requre a matr form of the dsspaton that takes nto account the characterstc structure, ]. Therefore, from one pont of vew, the success of upwnd schemes s attrbutable to ther ablty to naturally add the approprate amount and form of dsspaton to the appromaton by usng the characterstc structure of the equatons. Upwndng was eplctly ntroduced by Godunov n hs 959 landmark paper 3]. In that work, a frst-order accurate upwnd method was devsed for systems of hyperbolc partal dfferental equatons n frst-order form, ncludng a treatment of the nvscd Euler equatons. The key dea was to ncorporate an eact soluton of the Remann problem nto the numercal technque. Also recognzed n that work was the nablty for lnear schemes to both mantan monotone profles and at the same tme acheve hgher than frst order accuracy for smooth flows (Godunov s theorem). Snce the poneerng work of Godunov, there have been many etensons to the upwnd approach ncludng, for eample, the flu-corrected transport method 5], the pecewse-parabolc-method (PPM) 6], essentally-non-oscllatory (ENO) schemes 7, 8], dscontnuous Galerkn (DG) appromatons 9], and the weghted-essentally-non-oscllatory (WENO) class of methods ]. In ths paper, we demonstrate how to construct hgh-order accurate upwnd schemes for wave equatons n second-order form wthout transformng the equatons to a frst-order system. Ths s, to our knowledge, the frst systematc attempt to develop such schemes. The approach usually used to ncorporate upwnd dscretzatons nto hyperbolc PDEs n second-order form s to transform the equatons nto a frst-order system of equatons. For eample, the equatons of lnear elastcty, whch take the form of a system of second-order hyperbolc PDEs for the dsplacements, can be transformed nto a system of frst-order hyperbolc PDEs for velocty and stress ]. Ths s, of course, a vald and useful approach but there are a varety of reasons to consder solvng the second-order PDEs drectly. One advantage of drectly solvng the equatons n second-order form s that there are often sgnfcantly fewer dependent varables, and thus the schemes can be more effcent. For eample, for Mawell s equatons of electromagnetsm, the number of dependent varables can be reduced by a factor of two ]. For the equatons of lnear elastcty n three dmensons, the second-order form has just three dependent varables (for the dsplacement) compared to between nne and ffteen for the frst-order system (dependng on whch components of the dsplacement, velocty and stress are retaned). Another advantage of solvng the secondorder form of the equatons s the removal of ssues related to the equvalence, or lack thereof, between the frst- and second-order systems. In general, frst-order systems derved from second-order equatons Ths phlosophy s emboded by Greengard s aom; It never hurts to start by wrtng down the eact soluton to the problem 4].

3 can admt a wder class of solutons and thus constrants must be mposed to ensure that solutons to the frst-order system are also solutons to the second-order equatons (e.g. the Sant-Venant compatblty condtons for elastcty). For some systems, such as for the equatons of general relatvty, the number of constrant equatons can become nordnately large whch sgnfcantly complcates numercal appromaton. In addton, many systems occur naturally as PDEs n second-order form and from a phlosophcal perspectve there should be no partcular reason why upwnd style dscretzatons can t be derved n the natve form. Fnally we note that when solvng problems n multple space dmensons, t s often the case that a larger tme-step can be taken for the second-order form of the equatons compared to the frst-order form ]. An outlne of the remander of the manuscrpt follows. We begn, n Secton by developng a frst-order accurate upwnd scheme for the second-order wave equaton that uses the d Alembert soluton to eactly advance a pecewse smooth representaton of the soluton. Ths geometrc approach s analogous to Godunov s constructon for advancng a pecewse constant representaton of the soluton to the frst-order wave equaton. In Secton 3 we generalze ths prelmnary approach (whch reles on a global representaton of the eact soluton) to a method that only requres the soluton to local problems at cell faces. These localzed problems are a generalzaton of the Remann problem for the frst-order system. Ths localzed form s the key ngredent needed for etendng the scheme to multple space dmensons, hgh-order accuracy, varable coeffcents and systems of equatons (we leave varable coeffcents and systems to future work). From the localzed form we then derve hgh-order accurate space-tme schemes usng the Cauchy-Kowalewsk procedure. We choose to develop conservatve fnte-dfference appromatons, although fnte volume schemes are also possble. Schemes wth orders of accuracy one, two, four and s are developed and analyzed usng normal mode theory and modfed equatons. A second-order accurate hgh-resoluton scheme (based on the use of nonlnear lmters) s also developed. The deas are then generalzed to two space dmensons n Secton 4 where frst-order accurate, second-order accurate, hgh-resoluton and fourth-order accurate schemes are constructed and analyzed. The relatonshps between the modfed equatons, the soluton error and the accuracy requrements n terms of ponts per wavelength s dscussed n Secton 5. Numercal eamples are presented n Secton 6. Smooth solutons n one space dmenson (travelng sne wave) and two space dmensons (surface waves) are used to verfy that the ma-norm errors n the schemes converge at the epected rates. A top-hat problem, n both one and two space dmensons, s used to demonstrate the robust and hgh-qualty appromatons that result from the schemes when they are appled to dffcult problems wth dscontnuous ntal condtons. Secton 7 provdes concludng remarks whle the stencl coeffcents for the one-dmensonal fourth- and sth-order accurate schemes are gven n appendces.. Constructon of an upwnd scheme for the second-order wave equaton n one dmenson Godunov s method and ts etensons are well establshed for solvng frst-order systems of hyperbolc equatons. These upwnd-type methods use solutons to a set of Remann problems defned at cell faces of a computatonal grd to construct a dscrete appromaton to the soluton of a hyperbolc PDE. For lnear problems, the Remann soluton s defned n terms of the characterstcs of the hyperbolc operator. By usng the characterstc varables, these methods naturally ncorporate upwndng by tracng the nfluence of each characterstc varable along ts respectve space-tme path. It s useful to begn the dscusson by revewng a geometrcally motvated dervaton of a frst-order accurate Godunov scheme for the ntal value problem for the scalar frst-order wave equaton (FOWE), q t + c q =, < <, () q(, ) = q (). Here q = q(, t) and for smplcty we assume that c s constant wth c >. Introduce a unform grd wth cell centers = h and cell faces + = ( + )h, where h > s the grd spacng and =, ± ±,.... Let q n q(, t n ), be a grd functon representng the soluton at tme t n = n t, where t s the tme-step. At a gven tme t n we assume a known dscrete soluton at cell centers, q n and form the pecewse constant functon Q () = q n, for (, + ). 3

4 t t + t t C + q n+ q n q n q n + C + : = + + ct 3 + Fgure : Ths fgure llustrates the geometrc constructon of the frst-order Godunov scheme for the frst-order wave equaton q t + cq =. The pecewse constant functon that equals to q n on cell (, + ) s eactly advanced to to tme t + t. Ths s used to defne dscrete cell centered values, q n+, by ntegratng the soluton over the cell. The resultng scheme s the standard frst-order accurate upwnd scheme For τ = t t n, the eact soluton to () wth ntal condtons Q () can be found usng the method of characterstcs and s gven by Q(, τ) = q n, for cτ (, + ). The dscrete cell-centered soluton at the net tme-step t n+ s defned as the cell average of Q(, t), q n+ = + Q(, t)d. h For tme steps satsfyng t h /c ths constructon s llustrated n Fgure and gves the standard frst-order accurate upwnd scheme, q n+ = c tq n h + (h c t)q n ], = q n c t h (q n q n ). () Remark. Stablty analyss reveals that the upwnd scheme () s stable under the constrant t h /c. We now pursue a smlarly ntutve constructon appled to the one-dmensonal second-order wave equaton. Consder the ntal value problem for the second-order wave equaton (SOWE) n one-dmenson, u t = c u, < <, u u(, ) = u (), t (, ) = v(, ) = v (). In analogy wth elastcty, we call u = u(, t) the dsplacement and v = u/ t the velocty. An upwnd method to solve the SOWE can be constructed as follows. Let u n and v n, denote grd functons that are appromatons to u(, t n ) and v(, t n ), respectvely. Defne a pecewse smooth representaton of the dscrete soluton at tme t n (as shown n Fgure ) by U () = u n + h (u n + u n ), for (, + ), (3) V () = v n for ( + ). (4) Here U () s the contnuous and pecewse lnear functon that passes through the values (, u n ). V () s the pecewse constant functon that passes through the ponts (, v n ) and s constant on cells (, + ). The eact soluton correspondng to the ntal condtons U () and V () can be determned from the well known d Alembert soluton to the second-order wave equaton 3]. Lettng τ = t t n, ths soluton s 4

5 u n + u n v n u n v n v n Fgure : The appromate soluton for the second-order wave equaton n one-dmenson s represented by the dscrete grd functons u n (dsplacement) and v n (velocty) along wth the correspondng pecewse smooth functons shown n the fgure gven by U(, τ) = (U ( + cτ) + U ( cτ)) + c +cτ cτ V (ξ)dξ, (5) V (, τ) = t U(, τ) = c (U ( + cτ) U ( cτ)) + (V ( + cτ) V ( cτ)). (6) As shown n Fg. 3, the form of ths soluton depends on the representaton of U () and V () gven n t C C C C + C + C + t Fgure 3: The t dagram showng the characterstcs emanatng from the cell centers and cell faces. (3)-(4) and by the characterstcs, C + : d/dt = c, and C : d/dt = c, that emanate from the ponts ± and. To reconstruct the dscrete soluton at t n+ we compute cell averaged values of U(, t) and V (, t), u n+ = + U(ξ, t)dξ, v n+ = + V (ξ, t)dξ. h h These ntegrals can be evaluated eplctly usng the epressons (5)-(6) and the equatons (3)-(4) for U () and V. For c t/h < ths procedure gves the followng scheme, denoted by UWa (for upwnd scheme of order, verson a), u n+ = u n + t v n + ( c t + h 8 )D +D u n + c t 4 v n+ = v n + c t D + D u n + c t h D + D v n, (7) h D + D v n. (8) Here D + and D are the usual forward and backward dvded dfference operators defned by D + w = (w + w )/h and D w = (w w )/h. For future reference we also note that the undvded forward and backward dfference operators are defned by + w = w + w and w = w w. Equatons (7)-(8) defne a frst-order accurate upwnd scheme for the one-dmensonal second-order wave equaton. A normal mode analyss shows that the scheme s stable provded t Λ a h /c where Λ a = ( + 5)/4.89. The formal accuracy of ths scheme can be determned by computng the truncaton error. However, we proceed 5

6 further and derve the so-called modfed equatons from whch further propertes of the dscrete scheme can be gleaned. The modfed equaton s a contnuous PDE whose soluton descrbes the appromate behavor of the well resolved components of the dscrete equatons. The modfed equaton s derved by substtutng contnuous functons U(, t) and V (, t) nto the dscrete equatons (7)-(8) by settng u n = U(, t n ) and v n = V (, t n ), and epandng all terms n Taylor seres about the pont (, t) = (, t n ). Ths leads to the epanson U t = V t U t + c t V t = c U t V t U t 6 + ch 3 U t 3 V t 6 + c th V 4 + h 8 t 3 V t 3 + c h U + O(( t + h ) 3 ), (9) 4 U 4 + O(( t + h ) 3 ). () The tme dervatves on the rght hand sdes of (9)-() can be elmnated by dfferentatng (9)-() wth respect to tme and recursvely substtutng the result back nto the rght hand sdes of (9)-(). Ths leads to the followng modfed equatons satsfed by U and V U t V t = c U + ch 8λ = c V + ch 8λ ( ( 4λ + 4λ + ) 3 U t ( ( 4λ + 4λ + ) 3 V t ) + O(( t + h ) ), () ) + O(( t + h ) ), () where λ = c t/h s the CFL parameter. From the modfed equatons ()-() we see the scheme s formally frst-order accurate snce the frst correcton term s O(h ). Note that ths correcton term, (the term nvolvng 3 U/ t for the U equaton ()), s dsspatve n nature, and s a reflecton of the upwnd appromaton. The behavor of the upwnd scheme (7)-(8) s llustrated for a partcularly dffcult problem. The ntal condtons consst of a top-hat functon for the dsplacement and zero ntal velocty, ( u(, ) = H + ) ( H ), (3) 4 4 v(, ) =. (4) Here H() s the Heavsde functon defned by H() = for < and H() = for. The eact soluton can be found by appealng to d Alembert s soluton and s gven as u(, t) = ( H + ct + ) ( H + ct ) ( + H ct + ) ( H ct )], (5) v(, t) = c ( δ + ct + ) ( δ + ct ) ( δ ct + ) ( + δ ct )]. (6) Solvng ths problem numercally s dffcult snce the soluton for u does not have a contnuous frst dervatve, whle v s defned n terms of Drac delta functons. Fgure 4 shows the dscrete soluton for the dsplacement and velocty at a number of tmes, computed usng the scheme (7)-(8). For ths computaton we took c = and used N = grd ponts on the nterval, ] wth h = /(N ). The tme-step t was chosen to be a factor of.9 of the mamum allowed by stablty. For comparson we show, n Fgure 5, results from the UWa scheme, results from the UW scheme (47)- (48) (whch s derved n Secton 3. usng a more general procedure), and results computed usng the standard centered fnte dfference scheme to the SOWE gven by u n+ u n + un t = c D + D u n. (7) We denote the scheme (7) as the C scheme (for centered scheme of second-order accuracy). The velocty for the C scheme, as shown n ths fgure, s a( derved quantty computed usng a centered second-order fnte dfference operator n tme as v n = t u n+ u n ). Notce the qualtatve dfferences between the standard centered appromaton to the new upwnd appromatons. The centered scheme dsplays the usual hghly oscllatory phenomena near dscontnutes. The upwnd schemes are much smoother near the dscontnutes wth scheme UW havng a sharper resoluton than UWa. 6

7 .9.8 Second Order Wave Equaton: u, Top Hat (UWa), N= t=. t=.3 t=.5 5 Second Order Wave Equaton: v, Top Hat (UWa), N= t=. t=.3 t= u v Fgure 4: Results for the top-hat ntal condton of Secton 6. usng the prelmnary frst-order accurate upwnd scheme UWa. Results are shown at t =.,.3 and.5 for u (left) and v (rght). Notce that the locaton of the Drac delta functons n the eact soluton for v are ndcated by vertcal black lnes. Remark: We have found that care must be taken n mposng ntal condtons for the C scheme. For eample f the eact soluton at t = and t = t are used, convergence to the ntended weak soluton s not guaranteed even n an L sense. The appromaton presented n Fgure 5 was obtaned by applyng a second-order Runge-Kutta procedure from t = to t = t and then movng to the C scheme for subsequent tme-steppng. Ths choce for startng the computaton was prmarly made for convenence; other methods such as those based on Taylor epanson could also be consdered Second Order Wave Equaton: u, Top Hat t=.5, N= eact C UWa UW 3 Second Order Wave Equaton: v, Top Hat t=.5, N= eact C UWa UW Fgure 5: Results for the top hat ntal condton of Secton 6. usng the prelmnary frst-order scheme UWa, the C scheme, and the alternatve frst-order upwnd scheme UW (47)-(48). Left: dsplacements u. Rght: velocty v (note the vertcal scale). The eact (weak) soluton for the velocty s gven by four travelng delta functons whch are dsplayed as vertcal black lnes. Remark. The scheme of (7) and (8) s conservatve snce, lettng S(f ) = ν =µ f h denote a dscrete appromaton to the ntegral, then S(u n+ ) = S(u n ) + t S(v n )h + boundary terms, S(v n+ ) = S(v n ) + boundary terms. Remark. The step of formng the pecewse polynomal soluton U () and V () n (3)-(4) from the grd functons u n and vn s often called the reconstructon step 4]. There are many choces for ths reconstructon and we present just one. Ths one has the property that the d Alembert soluton gven by (5)-(6) has the same degree of smoothness as the ntal condtons;.e. U(, t) remans contnuous and pecewse lnear and V (, t) remans pecewse constant. In contrast, choosng a pecewse constant reconstructon for U (), 7

8 would result n a d Alembert soluton wth Drac delta functons n V (, t). It seems wse to avod such a sngular soluton n our dscrete representaton. In addton, for consstency t seems approprate to choose a reconstructon for U () that s one dervatve smoother than the reconstructon for V (). Ths s consstent wth the fact that v s a tme dervatve of u and that for the wave equaton the tme dervatve has the same degree of smoothness as a space dervatve. 3. A general constructon for hgh-order upwnd schemes the one-dmensonal case We now consder a more general development of upwnd schemes for the SOWE. Ths constructon s smlar n nature to the constructon of upwnd methods for the frst-order system usng solutons to Remann problems at cell faces rather than drect ntegraton of eact solutons as presented n Secton. Consder the ntal value problem for second order wave equaton n one space dmenson, u = Lu, Ω, (8) t L c, u(, ) = u (), u t (, ) = v (), where u = u(, t) s taken to be perodc n on the nterval Ω =, ] and where c s constant wth c >. We wrte (8) n the form of a conservaton law, ] ] ] u v = v c +. u t Introduce a unform Cartesan grd for Ω wth N + grd ponts. Let h = /N denote the grd spacng. Denote the grd cell centers by = ( + )h, =,,..., N and the grd vertces by = h, =,,..., N. We choose to pursue a so-called conservatve fnte-dfference approach by dscretzng the tme ntegrated equatons. One could alternately follow a fnte-volume approach and ntegrate the equatons over a space-tme cell. Ths choce s largely one of personal preference, and the mechancs of developng fnte-volume schemes should be clear from the followng dscusson. The dervaton of the scheme begns by ntegratng the equaton for the velocty n tme, v(, t) =v(, ) + c t u (, τ) dτ. (9) Ths epresson can be transformed nto a form that resembles a dscrete conservaton appromaton by frst wrtng u/ as the forward dvded dfference of the face centered flu functon f( h, t), u (, t) = D +f( h, t), () u f(, t) D (, t), () where the applcaton of D + to contnuous functons s defned to be D + f(, t) = (f( + h, t) f(, t))/h. The dfferental-dfference operator D s defned to satsfy the dentty ( w () = D + D w( h ) ), () for any suffcently smooth functon w and s formally gven by the epanson D w(, t) = j= α j h j j w (, t), (3) j = w(, t) h w 4 (, t) + 7h w (, t) 3h w (, t)

9 The coeffcents α j can be computed from the dentty ζ = snh(ζ/) j= α jζ j by equatng coeffcents of ζ n the Taylor seres epanson, followng the approach descrbed n 5, 6]. Usng () n (9) gves the followng equaton for v(, t) v(, t) =v(, ) + c D + t f( h ], τ) dτ. Ths equaton s ntegrated n tme to gve an eact equaton for u(, t), u(, t) = u(, ) + t v(, ) + c D + t τ f( h ], τ ) dτ dτ. Takng these ntegratons over a sngle tme-step leads to the followng conservaton form for the soluton at tme t n+ v(, t n+ ) = v(, t n ) + c td + F v ( h ], tn ), (4) u(, t n+ ) = u(, t n ) + t v(, t n ) + c t D + F u ( h ], tn ), (5) F v (, t n, t) t F u (, t n, t) t t f(, t n + τ) dτ, (6) t τ f(, t n + τ ) dτ dτ. (7) We emphasze that (4)-(7) s a formally eact dfferental-dfference equaton for the soluton. Dfferent numercal schemes are developed by consderng varous upwnd appromatons to f( h /, t n +τ) n (6)- (7). 3.. The upwnd flu for the second-order system One way to ncorporate upwndng nto the scheme (4)-(7) s to use -dervatves of the d Alembert soluton (5) to derve epressons for dervatves of u on the face n terms of the soluton at the prevous tme step t n, p p u(, tn + τ) = ( ) p p u( + cτ, tn ) + p p u( cτ, tn ) + ] p c p v( + cτ, tn ) p p v( cτ, tn ), for p =,,.... (8) Use of (8) n (4)-(5) could be used as a startng pont for a characterstc-based upwnd scheme for the SOWE. We do not, however, follow ths lne of appromaton here. Instead, we use another approach to ncorporatng upwndng that can be more easly ncorporated nto conservatve fnte dfference appromatons. Ths alternatve approach makes use of the fact that takng the lmt τ n (8) gves p p u(, t+) = ( ) p p u(+, t) + u(, t) p p + ] p p v(+, t) v(, t), for p =,,.... (9) c p p Here t+ denotes the lmt of t+ɛ as ɛ wth ɛ >. Smlarly, + and denote values of nfntesmally larger and smaller than, respectvely. Formula (9) s the key result that we wll use to derve upwnd schemes to any order of accuracy. It specfes how to defne an upwnd based flu on a face, gven left and rght appromatons to the spatal dervatves of the dsplacement and velocty on ether sde of the face. In partcular, the upwnd flu appromaton for the p th -dervatve of u s the average of the p th -dervatve of the left and rght dsplacement states, plus the recprocal of c tmes the dfference of the (p ) st -dervatve of the left and rght velocty states. Our assumed form of the dscrete soluton for a low order appromaton was prevously shown n Fgure and conssts of a pecewse lnear representaton for u and a pecewse constant representaton for v. For ths 9

10 form we note that the frst dervatve of u s contnuous at the cell faces whle v may jump. More generally we wll not assume any partcular pecewse smooth representaton for the soluton. Rather we proceed n the usual manner used to derve hgh-order fnte dfferent methods. We assume that the analytc soluton has as many contnuous dervatves as needed and formally epand the soluton n Taylor seres. In dervng our scheme we use a staggered approach 3 whch epands p u(, t)/ p n a truncated Taylor seres centered at faces whle epandng p v(, t)/ p n a Taylor seres about cell centers. As a result, the truncated seres appromaton to u and t s dervatves are contnuous at faces whle the seres appromatons to v and t s dervatves wll n general jump at faces. Wth ths assumed form of appromaton we can smplfy (9) (whch wll be appled at faces) by replacng the average of the left and rght appromatons to p u(, t) p n (9) wth a centered appromaton, p p u(, t+) = p p u(, t) + c Defnton The upwnd flu functon ˇf s defned to be ˇf( + h, tn + τ) D u ( +, tn + τ) + c ] p p v(+, t) v(, t), for p =,,.... (3) p p D v + ( +, tn + τ) D v ( +, tn + τ) where v and v + denote left and rght lmts, respectvely, to v on the face. ], (3) The upwnd flu functon (3) replaces f n the scheme (4)-(5) to defne an upwnd scheme. Upwnd schemes wth dfferent orders of accuracy are determned by appromatng terms n (3) wth the approprate accuracy. The Cauchy-Kowalewsk process, also known as the La-Wendroff procedure 7], s used to develop space-tme schemes usng data from only two tme levels. In ths approach, equaton (3) s epanded usng Taylor seres n space and tme, and the governng equatons are used to replace temporal by spatal dervatves. Specfcally, Taylor epansons of u/ about the face poston + and v about the cell center gve u ( +, τ m m u tn + τ) = m! t m ( +, tn ), (3) m= v( ± h m ( ) m, tn + τ) = n m! τ m n( ±h ) n m v(, t n ) t m n n. (33) m= n= From (33), left and rght based epressons for the velocty on the face are gven as v ( +, tn + τ) v( + h, tn + τ), (34) v + ( +, tn + τ) v( + h, tn + τ). (35) Note that the epanson determnng v orgnates from, whle that for v + orgnates from +. Tme dervatves can be converted to space dervatves usng the governng equaton u tt = Lu. Lettng, { L m/ u f m s even L m (u, v) L (m )/ v f m s odd, (36) t follows that m t u = L m (u, v) and m t v = L m+ (v, u). Usng (3)-(33) together wth (3) and defntons (34) and (35) n (3) gves, Proposton 3.. The space-tme epanson for the upwnd flu (3) s gven by { τ ˇf( +, tn + τ) = α j h j m j+ m! j+ L m(u, v)( +, tn ) + c m n= ( m n j= m= ) τ m n m! ( h ) } n n+j n+j ( )n L m n+ (v, u)( +, t n ) L m n+ (v, u)(, t n )]. (37) 3 The use of a staggered appromaton seems to be an mportant ngredent n the overall approach because t leads to more compact appromatons that are generally preferable for stablty and accuracy reasons.

11 The sngle and double tme ntegrals of the epanson for the upwnd flu (37) requred for (6)-(7) can be easly obtaned, whch leads to, Proposton 3.. Let {k p } denote a set of numbers {k p } p= where the form of each k p s to be specfed. Upwnd appromatons to the tme averaged flu functons (6)-(7) are gven by where F (τ, {k p }) = + c m n= F v ( +, t) F v + (t n, t) = F ( t, {k p = /(p + )}), (38) F u ( +, t) F u + (t n, t) = F ( t, {k p = /((p + )(p + ))}) (39) j= m= ( m n { α j h j km τ m m! j+ j+ L m(u, v)( +, tn ) ) km n τ m n ( h ) } n n+j m! n+j ( )n L m n+ (v, u)( +, t n ) L m n+ (v, u)(, t n )]. (4) Lettng L = c, the frst few terms n the epanson for F (τ, {k p }) are gven by u F (τ, {k p }) = k ( +, tn ) + k c +v(, t n ) + k τ v ( +, tn ) + k cτ u + (, t n ) k h c A v + (, t n ) + ( k c τ k h u 4 ) 3 3 ( +, tn ) + ( k cτ + k h 4 4c ) v + (, t n ) k cτh 3 u A + 3 (, t n ) + ( k 3c τ 3 k τh 6 4 ) 3 v 3 ( +, tn ) + ( k 3c 3 τ 3 + k cτh 4 u ) (, t n ) k cτ h 3 v A (, t n ) + O((h + t) 4 ). (4) Here A + s the forward averagng operator, A + u(, t n ) = (u( +, t n ) + u(, t n )). Remark. Note that most of the terms n (4) can be appromated n a natural way by usng a compact centered appromaton. For eample, a second-order accurate appromaton to u( +, tn )/ s the compact centered appromaton D + u(, t n ). There s, however, some choce n appromatng the un-centered terms nvolvng A +, such as v(, t n )/. For a second-order accurate appromaton to v(, t n )/ one could use ether a centered dfference appromaton (whch s not compact) or a forward or backward dfference appromaton. As a result, t s natural to consder lmtng these terms by choosng an approprate appromaton to the dervatve. For eample, one mght dynamcally choose the appromaton that has the smallest magntude. Lmters are dscussed further n Secton 3.3. From an mplementaton pont of vew t may be more effcent to use a quadrature rule, rather than ntegratng eactly n tme. In partcular, f one uses Gaussan quadrature wth quadrature ponts ξ j, j =,,..., M for the nterval, ], then appromatons to both F v ( +, tn, t) and F u ( +, tn, t) can be computed from M evaluatons of an appromaton to the flu functon (4) F j + F (ξ j t, {}), j =,,..., M, F v ( +, tn, t) F v + (t n, t) = F u ( +, tn, t) F u + (t n, t) = M j= M j= b j F j, (4) + d j F j. (43) + Here the weghts b j and d j are gven by the ntegrals of the nterpolaton bass functon φ j (τ), φ j (τ) = M k=,k j (τ ξ k) M k=,k j (ξ j ξ k ), b j = φ j (τ) dτ, d j = τ φ j (τ ) dτ dτ, (44)

12 and the quadrature ponts ξ k for an M-pont quadrature are the roots of the M th Legendre polynomal scaled to the nterval, ] (see 8] for detals). Note that for the specal case M =, the bass functon s gven by φ (τ) =. Snce Gaussan quadrature s eact for polynomals up to degree M and accurate to O( t M ), we can choose M = q for an appromaton of order q, q =,,.... Upwnd appromatons of dfferent orders of accuracy can now be defned by appromatng the terms n (4) to dfferent orders of accuracy and substtutng nto v n+ u n+ = v n + c t D + F v (t n, t) ], (45) = u n + t v n + c t D + F u (t n ) ]. (46) 3.. A frst-order accurate upwnd scheme A frst-order accurate upwnd scheme s defned by keepng the frst two terms n (4) and appromate these usng F ( t, {k p }) k u ( +, tn ) + k c +v(, t n ), F + ( t, k ) k D + u n + k c +v n. The scheme can be effcently evaluated usng Gaussan quadrature n tme wth M = and quadrature node ξ =, and b =, d =, to gve F = F (ξ t, k = ), g = t c D + F, v n+ = v n + g, u n+ = u n + t v n + t g. In ths case, usng eact ntegraton n tme leads to the same appromaton. Wrtten out n detal, the appromaton, denoted by UW, s gven by v n+ u n+ = v n + t c D + D u n + t ch D + D v n (47) = u n + t v n + t c D + D u + t 4 ch D + D v n. (48) Ths scheme (47)-(48) dffers slghtly from the UWa scheme (7)-(8) derved n Secton. The tme step restrcton s larger as noted below, and ths scheme s generally less dsspatve (see Fgure 5). The modfed equatons for the scheme (47)-(48) are gven by U t = V + h λ V + O(( t + h ) 3 ), V t = c U + ch ( V λ) + c h ( 3λ + λ ) 4 U 4 + O(( t + h ) 3 ), where, as before, λ = c t/h. Ths system of equatons for U and V can be wrtten as decoupled equatons by dfferentatng wth respect to tme and substtuton. Ths results n the set of decoupled modfed equatons U t V t = c U + ch ( λ) 3 U t + c h ( 3λ + λ ) 4 U 4 + O(( t + h ) 3 ), (49) = c V + ch ( λ) 3 V t + c h ( 3λ + λ ) 4 V 4 + O(( t + h ) 3 ). (5) Note that the modfed equatons for U and V are the same to the order gven. From equatons (49) and (5) we see that (for λ ) the appromaton s frst order accurate n space, O(h ), and frst order accurate n tme, O( t) (note the defnton of λ). The terms proportonal to h on the rght hand sdes of (5) and (49) represent dsspaton terms whch wll damp the soluton (hgher-frequences wll be damped more strongly)

13 provded λ (, ). The terms proportonal to h on the rght hand sdes add dsperson to the equatons. The dampng and dsperson propertes can be seen by Fourer transformng the equatons. Remark: The modfed equatons (5)-(49) ndcate that at λ = the scheme actually becomes hgher than frst-order accurate. In fact scheme UW has an nfnte order of accuracy (spectrally accurate) for λ = (see below). Lemma 3.3. The UW scheme (47)-(48) s frst-order accurate and stable provded t h c. (5) Proof The accuracy and stablty of the scheme can be determned through a normal mode analyss 9]. To ths end we seek solutons to (47)-(48) of the form ] û ] u n j = z n e πk j, ˆv v n j where z s a comple valued amplfcaton factor, and û and ˆv are Fourer coeffcents. The wave number k takes the dscrete values k =, ±, ±,... ± N/ (n the case N s an even nteger). Substtuton nto (47) and (48) leads to an egenvalue problem for z wth egenvector û ˆv] T, z λ ˆξ t( λ ˆξ ] û ] 4 ) t λ ˆξ z λ ˆξ = ˆv Here ξ = πk/n, wth ξ π, π], whle ˆξ = 4 sn (ξ/) s the Fourer symbol of +. Nontrval solutons to ths problem est when the determnant s zero. Ths leads to a quadratc equaton for z wth roots z = z ± where Epandng (5) for small ξ yelds ]. z ± = b ± b a, (5) a = λ ˆξ, b = 4 (a + λˆξ ). (53) z ± = ± λξ 4 λ( + λ)ξ + O(ξ 3 ) = e ±λξ + O(ξ ). Snce z = ep(±λξ) corresponds to the eact soluton to the contnuous problem, t follows that z ± are second-order accurate appromatons to ep(±λξ) and thus the overall dscrete soluton s frst-order accurate at tmes of order (.e. after O(/ t) tme-steps). Stablty requres that the roots satsfy z ±. Contour plots of the magntude of the two roots as a functon of ξ π, π] and λ,.5] are shown n Fgure 6. The scheme s seen to be stable for λ. Indeed, assumng λ t s easly shown that z ± = ˆξ λ( λ), and thus z ± for λ snce ˆξ 4. The tme-step restrcton (5) follows. For a fully algebrac proof of ths result see the proof for the two-dmensonal scheme, Lemma 4., whch ncludes the one-dmensonal result as a specal case. Remark: For λ = the roots z ± n (5) reduce to z ± = cos(ξ) ± sn(ξ) = e ±ξ. (54) Ths shows that scheme UW has spectral accuracy when λ =. The only error comes from representng the ntal condtons as a fnte dscrete Fourer seres. 3

14 .5 Frst order scheme UW: z +.5 Frst order scheme UW: z λ λ ξ 3 3 ξ Fgure 6: Regon of stablty for the frst-order accurate scheme UW: contours of the magntude of the two roots z ±, as a functon of the CFL parameter λ and the Fourer parameter ξ. The contour level s shown as a thck black lne. The scheme s unstable where the magntude of ether root eceeds A second-order accurate scheme and a hgh-resoluton scheme A second-order accurate upwnd scheme s defned by keepng the frst fve terms n (4) F ( t, {k p }) k u ( +, tn ) + k c +v(, t n ) + k t v ( +, tn ) + k c t u + (, t n ) k h c A + and usng the appromaton v (, t n ), (55) F + ( t, k, k ) k D + u n + k c +v n + k td + v n + k c t + D + D u n k h c A +D v n. The scheme usng eact tme ntegraton s then defned by F v + (t n, t) F + ( t, k =, k = ), F u + (t n, t) F + ( t, k =, k = 6 ). The scheme usng Gaussan quadrature n tme wth M = and quadrature node ξ =, and b =, d =, s defned by F + = F + (ξ t, k =, k = ), F v + (t n, t) = F +, F u + (t n, t) = F +. The Gaussan quadrature scheme can be effcently evaluated as g = t c D + F, v n+ = v n + g, u n+ = u n + t v n + t g. Wrtng out the appromatons n detal leads to the scheme (denoted by UW) u n+ v n+ = v n + t c D + D u n, + t c D + D v n + t 4 c3 h (D + D ) u n t 8 ch3 (D + D ) v n (56) = u n + t v n + t c D + D u + t3 4 c D + D v n + t3 8 c3 h (D + D ) u n t 6 ch3 (D + D ) v n. (57) 4

15 As for the case of frst-order systems, slope lmters can be ntroduced nto the appromaton to reduce oscllatons near dscontnutes n the soluton or ts dervatves. The lmted scheme wll be referred to as a hgh-resoluton scheme followng Harten ]. The hgh-resoluton varant of the second-order scheme (56)- (57) uses a lmted appromaton to the term nvolvng A + v(, t n )/ n (55) v A + (, t n ) SL (D + v n +, D v+) n + S L (D + v n, D v n ) ]. (58) Here S L s a slope-lmter functon that n ths case attempts to lmt the magntude of the dscrete appromaton that s used for v/. There are many possble choces for lmter functons. A common choce, the one used here, s the mnmum modulus functon defned by a f a < b and ab > MnMod(a, b) = b f a b and ab > (59) f ab. The hgh-resoluton scheme, denoted by HR, uses S L (a, b) = MnMod(a, b) n (58). Note that the choce S L (a, b) = (a + b) results n the orgnal centered appromaton. Fgure 7 compares the soluton from the frst-order (UW), second-order (UW) and second-order hghresoluton (HR) schemes for the top-hat problem (3)-(4). The tme step was chosen for each scheme to be.9 tmes the mamum allowable tme-step for that scheme (.e. a CFL number equal to.9). The hghresoluton scheme HR s seen to have sgnfcantly smaller undershoots and over-shoots compared to UW. The frst-order scheme UW gves the best soluton n ths case although ths s somewhat fortutous snce the UW scheme becomes spectrally accurate at λ =. Later n Secton 6. the dscrete L -norm errors for ths problem are shown (Fgure 8). Whle scheme UW convergences more slowly than the other schemes, the L -norm errors are smaller than the UW and HR schemes for a wde range of h. These advantages of the UW scheme dsappear, however, n two or three space dmensons snce the stable tme-step wll lead to much smaller values of λ beng used. Remark: We note, as shown by the results n Fgure 7, that the lmted hgh-resoluton scheme HR does not yeld the same type of monotone behavour ehbted by the frst-order upwnd scheme UW. We have so far not made any attempt to analyse or optmze the use of lmters for these new upwnd schemes. We envson that the nvestgaton of more sophstcated lmtng procedures, ncludng lmtng for the fourthand hgher-order accurate schemes, s an nterestng avenue for future research. Remark: It mght seem that the hgh-resoluton scheme should reman second-order accurate for smooth solutons. However, as shown n Secton 6., whle the ma-norm errors n u converge at order, the manorm errors for v are degraded to a convergence rate of 4/3. Ths degradaton can be traced to the lack of smoothness n the MnMod lmter. In order to understand the accuracy of the scheme (56)-(57), the leadng terms of the modfed equaton are derved through Taylor epanson. Followng the steps dscussed prevously for the frst-order accurate scheme leads to the followng system of modfed equatons U t V t = c U + c h M (λ) 4 U 4 + ch3 C (λ) 5 U t 4 + O(( t + h ) 4 ), (6) = c V + c h M (λ) 4 V 4 + ch3 C (λ) 5 V t 4 + O(( t + h ) 4 ), (6) M (λ) = ( + 3λ λ ), C (λ) = 8 ( + λ ). (6) The modfed equatons (6)- (6) are second order accurate appromatons to the true equatons n both space and tme (.e. O( ) for t = λ /c). The frst error term n the modfed equatons s a dsperson term and the second s a dsspaton term (provded C (λ) ). Lemma 3.4. The scheme UW (56)-(57) s second-order accurate and stable provded where Λ = t Λ h c, (63) 5

16 .6.5 Second Order Wave Equaton: u, Top Hat t=.5, N= eact UW UW HR Fgure 7: Results for the top hat ntal condton of (3)-(4) usng the frst-order accurate scheme UW, the secondorder accurate scheme UW, and second-order accurate hgh-resoluton scheme, HR. Soluton u at t =.5 usng N = grd ponts. Proof The accuracy and stablty for the second-order accurate scheme follows the same normal mode approach as for the frst-order scheme. Smlar steps lead to the egenvalue problem z γ + 8 η t( 4 γ 6 ζ) ] û ] ] t (γ 4 η) z γ 8 ζ =. ˆv where The two egenvalues z = z ± are gven by Epandng (64) for small ξ gves γ λ ˆξ, δ λˆξ, η δγ, ζ λˆξ 4. z ± = b ± b a, (64) a = γ( 4 δ), b = (γ 8 η + ζ). (65) 8 z ± = ± λξ (λξ) ± () 3 λ 4 ( + 3λ λ )ξ 3 + O(ξ 4 ) = e ±λξ + O(ξ 3 ), (66) and thus the soluton s second-order accurate. Note that the coeffcent of the ξ 3 term n (66) s closely related to the coeffcent M (λ) of the leadng error term n the modfed equaton (6). Ths s not a concdence and the general relatonshp s dscussed later n Secton 5 (see Fgure 3). Contour plots of the magntude of the two roots are shown n Fgure 8. The fgures show that the scheme frst becomes unstable on ξ = ±π. These are the most oscllatory plus-mnus modes. For ξ = ±π the stablty condton reduces to λ + λ, whch leads to the tme-step restrcton (63). For a fully algebrac proof of ths result see the proof for the two-dmensonal scheme, Lemma 4.3, whch ncludes the one-dmensonal result as a specal case. 6

17 .5 Second order scheme UW: z +.5 Second order scheme UW: z λ λ ξ 3 3 ξ Fgure 8: Regon of stablty for the second-order accurate scheme UW: contours of the magntude of the two roots z ± as a functon of the CFL parameter λ and the Fourer parameter ξ. The contour level s shown as a thck black lne. The scheme s stable for λ The modes whch determne the stablty bound are at ξ = ±π A fourth-order accurate upwnd scheme For fourth-order accuracy, (4) s appromated usng u F ( t, {k p }) k ( +, tn ) + k c +v(, t n ) + k t v ( +, tn ) + k c t u + (, t n ) k h + ( k c t + ( k 3c t 3 6 k h u 4 ) 3 3 ( +, tn ) + ( k c t + k h 4 k th ) 3 v 4 3 ( +, tn ) + ( k 3c t 3 c A v + (, t n ) 4c ) v + (, t n ) k c th + k c th 4 u ) (, t n ) A + 3 u 3 (, t n ) k c t h 3 v A (, t n ), (67) wth the dscrete appromaton Here F + ( t, k, k, k, k 3 ) k D + ( h 4 D +D )u n + k c +v n + k td + ( h 4 D +D )v n + k c t + D () 4h un k h c A +D ( h 6 D +D )v n + ( k c t k h 4 )D +D u n + ( k c t 4 + ( k 3c t k h 4c ) +D () 4h vn k c th A + D D + D u n k th )D 4 +D v n + ( k 3c 3 t 3 + k c th ) + (D + D ) u n 4 k c t h A + D D + D v n. (68) 4 D () 4h = D +D ( h D +D ), defnes a cell-centered fourth-order accurate appromaton to /. Gaussan quadrature n tme wth M = uses the quadrature nodes ξ = ( 3) and ξ = ( + 3) along wth b = b =, d = 4 + 3, 7

18 d = 4 3. The Gaussan quadrature scheme can be evaluated usng F + = F + t, {}), F + = F + t, {}), (69) F v + = b F + + b F +, F u + = d F + + d F +, (7) v n+ = v n + c td + F v + (7) u n+ = u n + t v n + c t D + F u +. (7) Wrtten out n detal, the fourth-order accurate upwnd scheme s (denoted by UW4) v n+ = v n + c t G (k =, k =, k = 3, k 4 = ), 4 (73) u n+ = u n + tv n + c t G (k =, k = 6, k =, k 3 = ), 8 (74) G (k p ) = k D () 4h un + k td () 4h vn + k c t (D + D ) u + k 3 6 c t 3 (D + D ) v ( 5k + 88c h5 k ) ( c t h 3 (D + D ) 3 v + k 8 c th3 + k ) 3 c3 t 3 h (D + D ) 3 u. (75) The stencl coeffcents for ths scheme are gven n Append A.. Lemma 3.5. The scheme UW4 (73)-(75) s fourth-order accurate and stable provded t Λ 4 h c, where Λ 4 s gven by the smallest real postve root of the polynomal equaton and s gven appromately by Λ λ 5 4λ 4 λ 3 + λ + 5 =, (76) Proof To determne the accuracy and stablty of ths fourth order scheme, a normal mode analyss s performed as before, gvng the egenvalue problem z λ β 4 + λ4 ˆξ λ3 ( 48 6 λ )ˆξ 6 t( λ 6 β 4 + λ4 ˆξ ] û λ 36 ( ] ] 4 λ )ˆξ 6 ) t ( λ β 4 + λ4 ˆξ 4 6 ) + λ3 ( 6 48 λ )ˆξ 6 ) z λ β 4 + λ4 ˆξ λ 36 ( =, λ )ˆξ 6 ˆv where β 4 = ˆξ ( + ˆξ ). The determnant condton s of the same form as before and there wll be two roots z ±. Epandng the roots for small ξ we fnd that z ± = ± λξ + (λξ) ± 3! (λξ)3 + 4! (λξ)4 ± λ M 4(λ)ξ 5 + O(ξ 6 ), = e ±λξ + O(ξ 5 ), showng that the soluton s fourth-order accurate (the coeffcent M 4 (λ) s gven below (77)). Contour plots of the magntude of the two roots are shown n Fgure 9. Interestngly, the modes whch determne the stablty bound occur near the lowest frequency mode, ξ = (The mode ξ = s tself stable wth ampltude ). For ξ z ± = P 6 (λ)ξ 6 + O(ξ 8 ), where P 6 (λ) s a polynomal n λ. The requrement z ± requres P 6 (λ). Ths leads to the stablty lmt beng gven by the smallest real postve root of the polynomal equaton (76), resultng n a stablty bound of Λ

19 The modfed equaton for the fourth order scheme s U t = c U + c h 4 M 4 (λ) 6 U 6 + ch5 C 4 (λ) 7 U t 6 + O(( t + h ) 6 ), M 4 (λ) = 6 ( 4 35λ + 6λ + 45λ 3 6λ 4 ), (77) C 4 (λ) = 864 (5 + λ λ 3 4λ 4 + 4λ 5 ), where V satsfes the same equaton as U. As was the case for the modfed equatons (6)-(6) for the second-order accurate scheme UW, the leadng error term n the modfed equatons for scheme UW4 s a dspersve term whle the followng term provdes the dsspaton (assumng C 4 (λ) )..5 Fourth order scheme UW4: z +.5 Fourth order scheme UW4: z λ λ ξ 3 3 ξ Fgure 9: Regon of stablty for the fourth-order accurate scheme UW4: contours of the magntude of the two roots z ± as functons of the CFL parameter λ and the Fourer parameter ξ. The contour level s shown as a thck black lne. The modes whch determne the stablty bound are near ξ = for λ A sth-order accurate upwnd scheme The sth-order accurate upwnd scheme UW6 s constructed followng the same procedures as outlned for the lower order schemes. We do not wrte down all the detals here snce the process for developng the scheme should be clear from the prevous cases. The stencl coeffcents for the UW6 scheme are gven n Append A.. Lemma 3.6. The scheme UW6 s sth-order accurate and stable provded where Λ 6 s gven by the smallest real postve root of the polynomal equaton t Λ 6 h c, (78) λ λ 99 λ λ λ λ 6 and s gven appromately by Λ λ λ λ λ + 65 λ 8 λ =, (79) Proof The accuracy and stablty analyss of the UW6 scheme proceeds as before. The roots z ± satsfy z ± = e ±λξ λ M 6(λ)ξ 7 + O(ξ 8 ), and thus the soluton s sth-order accurate (the coeffcent M 6 (λ) s gven below (8)). Fgure shows contours of the magntudes of the two roots z ± from whch t s seen that the hgh-frequency modes determne the stablty bound. Eamnaton of the magntude of the roots z ± for ξ = ±π leads to the the stablty condton (78). 9

20 The modfed equaton for scheme UW6 s gven by (V satsfes the same equaton as U) U t = c U c h 6 M 6 (λ) 8 U 8 + ch7 C 6 (λ) 9 U t 8 + O(( t + h ) 8 ), M 6 (λ) = 8 (8 + 47λ 49λ 7λ 3 + λ 4 97λ 5 ), (8) C 6 (λ) = 864 ( 3 4λ + λ λ 4 λ 5 53λ 6 + 6λ 7 ). As for schemes UW and UW4, the error terms n the modfed equatons for the sth-order accurate scheme begn wth a dspersve term, followed by a dsspaton term. A summary of the form of the modfed equatons for the dfferent schemes s gven n the Secton 5 (see Fgure 3)..5 Sth order scheme UW6: z +.5 Sth order scheme UW6: z λ λ ξ 3 3 ξ Fgure : Regon of stablty for the sth-order accurate scheme UW6: contours of the magntude of the two roots z ± as functons of the CFL parameter λ and the Fourer parameter ξ. The contour level s shown as a thck black lne. The scheme frst becomes unstable on ξ = ±π for λ Upwnd schemes n two space dmensons The one-dmensonal upwnd schemes devsed n prevous sectons can be etended to more space dmensons and n ths secton we show how to construct schemes n two space dmensons. The generalzaton to three space dmensons should be clear gven the subsequent dscusson. The approach n two-dmensons wll closely follow the developments of Secton 3. The two-dmensonal wave equaton n second-order form wll be ntegrated n tme and put nto the form of a conservaton equaton for the dsplacement, u, and velocty, v. Upwnd flu functons n the - and y-drectons wll be defned to be of the same form as the one-dmensonal upwnd flu functon (3). Space-tme schemes wll then be defned usng the Cauchy- Kowalewsk procedure. Two-dmensonal schemes wth orders of accuracy up to four wll be constructed and analyzed. Consder the ntal-value problem for the followng second order wave equaton n two space dmensons u = Lu, Ω, (8) t L c + c y y, (8) u(, ) = u (), u t (, ) = v (), where = (, y), and u = u(, y, t) = u(, t) s the dsplacement. Let Ω =, ] be the unt square and take u to be perodc n both and y drectons. Furthermore, assume that c and c y are constant, wth c > and c y >. As for the one-dmensonal case, we ntroduce the velocty v(, t) = u t (, t) and rewrte (8) as the equvalent system u v ] t = c u ] + ] c yu y y v + ]. (83)

21 4.. Dscretzaton n two space dmensons Dscrete appromatons to (8) wll be defned based on conservatve fnte dfferences. Introduce a unform Cartesan grd on Ω wth (N + ) (N y + ) grd ponts. Let h = /N and h y = /N y denote the grd spacngs. Denote the grd cell centers by,j = (, y j ) = (( + )h, (j + )h y), =,,..., N, j =,,..., N y and the grd vertces by,j = (h, jh y ), =,,..., N, j =,,..., N y. For compactness we also wrte u = u,j and =,j where = (, j) s a mult-nde. Let u n u(, t n ) and v n v(, t n ) denote dscrete appromatons to the dsplacement and velocty. Let D +, D, D +y, D y, denote the forward, and backward dvded dfference operators n the and y drectons, and +,, +y, and y the related undvded dfference operators. Let A + and A +y denote the forward averagng operators n the - and y-drectons. For eample, D + w = (w +,j w,j )/h, D y w = (w,j w,j )/h y, +y w = (w,j+ w,j ) and A +y w = (w,j+ + w,j )/. Followng the procedure developed n Secton 3, equaton (83) s ntegrated n tme over a tme step t to obtan equatons for v(, t n+ ) and u(, t n+ ) n terms of the soluton at tme t n. The contnuous spatal dervatves are transformed nto the form of a dscrete conservaton law. Ths leads to the followng formally eact dfferental-dfference equatons for the soluton, where v(, t n+ ) = v(, t n ) + c t D + F v ( h, y, tn ) + c y t D +y F v y (, y h y, tn ), (84) u(, t n+ ) = u(, t n ) + t v(, t n ) + c t D + F u ( h, y, tn ) + c y t D +y F u y (, y h y, tn ), (85) F v (, t n ) = t F u (, t n ) = t t t τ ˇf (, t n + τ) dτ, and where the upwnd flu functons are gven by ˇf ( + h, y, tn + τ) D u ( + h F v y (, t n ) = t t ˇf (, t n + τ ) dτ dτ, F u y (, t n ) = t + c ˇf y (, y + h y, tn + τ) D y u y (, y + h y + c y ˇf y (, t n + τ) dτ, (86) t τ ˇf y (, t n + τ ) dτ dτ, (87), y, tn + τ) D v + ( + h, y, tn + τ) D v ( + h ], y, tn + τ),, tn + τ) D y v +y (, y + h y, tn + τ) D y v y (, y + h ] y, tn + τ). Here the dfferental-dfference operator D y s defned n a smlar way to D but for dfferences and dervatves n the y-drecton nstead of the -drecton. The left and rght based velocty states are defned by v + ( + h, y, tn + τ) = D v( + h, y, tn + τ), v ( + h, y, tn + τ) = D v( + h, y, tn + τ), v +y (, y j + h y, tn + τ) = D y v(, y j+ h y, tn + τ), v y (, y j + h y, tn + τ) = D y v(, y j + h y, tn + τ). Followng the developments n Secton 3. we are led to Proposton 4.. Upwnd appromatons to the tme averaged flu functons (86)-(87) are gven by F( v + j, t n ) F v +,j(tn, t) = F ( t, {/(p + )}), (88) Fy v (, y j+ ) F v,j+ (t n, t) = F y ( t, {/(p + )}), (89) F u ( + j, t n ) F v +,j(tn, t) = F ( t, {/((p + )(p + ))}), (9) Fy u (, y j+ ) F u,j+ (t n, t) = F y ( t, {/((p + )(p + ))}), (9)

22 where F (τ, {k p }) = n= j= m= { α j h j km τ m j+ L m (u, v)( m! +, y j, t n ) + m ( ) m km n τ m n ( h ) } n n+j ( ) n L m n+ (v, u)( +, y j, t n ) L m n+ (v, u)(, t n )], (9) c n m! n= { F y (τ, {k p }) = α j h j km τ m y y j+ L m (u, v)(, y m! j+, tn ) j= m= + m ( ) m km n τ m n ( hy ) } n n+j y ( ) n L m n+ (v, u)(, y j+, t n ) L m n+ (v, u)(, t n )]. (93) c y n m! Note that the operator L m (u, v) n (9)-(93) s defned by (36), wth L gven by (8). The frst few terms n F (τ, {k p }) and F y (τ, {k p }) are gven by F (τ, {k p }) = k u ( +, y j, t n ) + k c + v(, t n ) + k τ v ( +, y j, t n ) + k τ c + Lu(, t n ) k h c A + v (, t n ) + O((h + t) ), (94) F y (τ, {k p }) = k u y (, y j+, tn ) + k c y +y v(, t n ) + k τ v y (, y j+, tn ) + k τ c y +y Lu(, t n ) k h y c y A +y v y (, t n ) + O((h y + t) ). (95) As n one-dmenson, Gaussan quadrature (44) can be used nstead of eact ntegraton n tme whch results n the appromatons F v ( +, y j, t n ) F v +,j(tn, t) = Fy u (, y j+, tn ) F u,j+ (t n, t) = M b j F (ξ j t, {}), j= M d j F y (ξ j t, {}), wth smlar epressons for F v y (, y j+, tn ) and F u ( +, y j, t n ). The fully dscrete scheme s then gven by v n+ = v n + c t D + F v,j(tn, t) ] + c y t D +y F v,j (t n, t) ], (96) u n+ = u n + t v n + c t D + F u,j(tn, t) ] + c y t D +y F u,j (t n, t) ]. (97) 4.. A frst-order accurate scheme n two dmensons A frst-order accurate upwnd scheme s defned usng wth the appromatons j= F ( t, {k p }) k u ( +, y j, t n ) + k c + v(, t n ), F y ( t, {k p }) k u y (, y j+, tn ) + k c y +y v(, t n ), F +/,j ( t, k ) k D + u n + k c + v n, F y,j+/ ( t, k ) k D +y u n + k c y +y v n.

23 The scheme can be effcently evaluated usng Gaussan quadrature n tme wth M = and quadrature node ξ = and b =, d =, to gve F, = F,j,j(ξ t, k = ), F y, = F y (ξ,j,j t, k = ), ( ) g = t c D + F, +,j c yd +y F y,,,j v n+ = v n + g, u n+ = u n + t v n + t g. Wrtten out n detal, the scheme s (denoted by UW-D) v n+ =v n + t L h u n + t (c h D + D + c y h y D +y D y ) v n, (98) u n+ =u n + t v n + t L hu n + t 4 (c h D + D + c y h y D +y D y ) v n, (99) L h c D + D + c yd +y D y, () where L h s a second order accurate appromaton to the operator L. Lemma 4.. The scheme UW-D (98)-(99) s frst-order accurate and stable provded t c + c ] y. () h h y Proof A normal mode stablty analyss of the scheme s carred out by seekng solutons of the form ] û ] u n µ,ν = z n e π(kµ+kyyν), ˆv v n µ,ν Ths ansatz leads to the egenvalue problem z λ ˆξ λ y ˆξ y t( λ ˆξ 4 λy ˆξ ] û ] 4 y) t (λ ˆξ + λ ˆξ y y) z λ ˆξ λy ˆξ = ˆv y Here ξ = πk /N wth ξ π, π], ξ y = πk y /N y wth ξ y π, π], ˆξ = 4 sn (ξ /) and ˆξ y = 4 sn (ξ y /). The two egenvalues are z ± = b ± b a, a = λ ˆξ + λ y ˆξ y, b = 4 (a + λ ˆξ + λ y ˆξ y ). Note that z ± should appromate the eact soluton e ±ω t where ω t = small ξ and ξ y gves ]. λ ξ + λ yξ y. Epandng z ± for z ± = ± λ ξ + λ yξy λ 4 (λ + )ξ λ y 4 (λ y + )ξy + O((ξ + ξ y ) 3 ), = e ±ω t + O((ξ + ξ y ) ), Ths shows that the scheme s frst-order accurate. To determne the stablty bounds consder the followng two cases. Case : when b < a, z ± = + a b. Stablty requres z ±, or λ ( λ )ˆξ + λ y ( λ y )ˆξ y, and ths s satsfed provded λ and λ y. Case : when b a, z ± mples b ± b a 3

24 whch leads to the condton 4 4b + a or Snce ˆξ 4 and ˆξ y 4 t follows that for stablty λ ˆξ + λ y ˆξ y 4. λ + λ y, whch leads to the tme step restrcton (). Ths condton s more restrctve than that for Case and thus s the requred stablty condton. Note that as n one-dmenson, the most unstable modes are the hghest frequency modes, ξ = ±π and ξ y = ±π. The modfed equaton for the two-dmensonal frst-order scheme UW-D s found to be U t = U c + U c y y + c h ( λ ) 3 U t + c yh y ( λ y) 3 U t y + c h ( 3λ + λ ) 4 U 4 + c yh y ( 3λ y + λ y) 4 U y 4 + h h y c c y 4 U (4λ λ y 3λ 3λ y ) y + O(( t + h + h y ) 3 ), () wth V satsfyng the same equaton. The frst set of error terms n () (those wth coeffcents proportonal to h or h y ) are the dsspaton terms (provded λ and λ y ). The subsequent terms (wth coeffcents proportonal to h, h h y or h y) are dsperson terms A second-order accurate scheme and a hgh-resoluton scheme n two dmensons Second-order accurate and hgh-resoluton schemes n two dmensons can be defned by keepng the frst fve terms n (94) and (95), F ( t, {k p }) k u ( +, y j, t n ) + k c + v(, t n ) + k t v ( +, y j, t n ) + k t + Lu(, t n ) k h v A + c c (, t n ), F y u ( t, {k p }) k y (, y j+, tn ) + k +y v(, t n ) c y and appromatng these wth + k t v y (, y j+, tn ) + k t c y F +/,j ( t, k, k ) k D + u n + k c + v n +y Lu(, t n ) k h y c y A +y v y (, t n ), + k td + v n + k t + L h u n k h A + S L (D v n, D + v n ), c c F y,j+/ ( t, k, k ) k D +y u n + k c y +y v n + k td +y v n + k t +y L h u n k h y A +y S L (D y v n, D +y v n ). c y c y The unlmted second-order accurate scheme (denoted by UW-D) uses S L (a, b) = (a + b)/ whle the hghresoluton scheme (denoted by HR-D) uses S L (a, b) = MnMod(a, b) (equaton 59). The appromaton can be effcently evaluated usng Gaussan quadrature n tme wth M = and quadrature node ξ = (b =, d = ) usng = F,j,j(ξ t, {}), F y, = F y (ξ,j,j t, {}), (3) ( ) g = t c D + F,,j c yd +y F y,,j, (4) v n+ = v n + g, (5) u n+ = u n + t v n + t g. (6) F, 4

25 Lemma 4.3. The unlmted scheme UW-D (3)-(6) (wth S L (a, b) = (a + b)/) s second-order accurate and stable provded ] ( c + c y ) 5 + c cy ( c + c y) t c + c, (7) y where c = c /h and c y = c y /h y. Proof For the second-order accurate unlmted scheme, normal mode analyss leads to the egenvalue problem, z γ + 8 η t( 4 γ 6 ζ) ] û ] ] t (γ 4 η) z γ 8 ζ =, ˆv where The two egenvalues z = z ± are gven by γ λ ˆξ + λ y ˆξ y, δ λ ˆξ + λ y ˆξ y, η δγ, ζ λ ˆξ4 + λ y ˆξ4 y. z ± = b ± b a, a = γ( 4 δ), b = (γ 8 η + 8 ζ). Epandng z ± for small ξ and ξ y we obtan z ± = ± W + (W ) + O((ξ + ξ y ) 3 ), = e ±ω t + O((ξ + ξ y ) 3 ), where W = λ ξ + λ yξy and ω t = W. Ths shows that the scheme s second-order accurate. To analyze the stablty of the scheme consder the two cases. Case : f b < a then z ± = + a b = (η + ζ), 8 and the scheme s always stable snce η and ζ. Case : f b a then whch leads to the condton 4 4b + a or b ± b a γ + 4 ζ 4. The left hand sde of ths last epresson s mamzed when ξ = ±π and ξ y = ±π and ths leads to the stablty condton λ + λ y + λ + λ y. In terms of a condton on t ths mples (7). The modfed equaton for U for the scheme UW-D s determned to be (the equaton for V s the same as that for U), U t = U c + U c y y + c h M (λ ) 4 U 4 + c yh y M (λ y ) 4 U y 4 + c c y h h y 4 U (3λ + 3λ y λ λ y ) y (8) + c h 3 C (λ ) 5 U t 4 + c yh 3 y C (λ y ) 5 U t y 4 λ λ y h h y 5 U (c y h + c h y ) 8 t y (9) + O(( t + h + h y ) 4 ), where M and C are gven by (6). The frst set of error terms (8) represent dspersve terms whle the second set (9) are the ones that add dsspaton. 5

26 4.4. A fourth-order accurate scheme n two dmensons A fourth-order accurate upwnd scheme s derved followng the process developed n prevous sectons. These are a straghtforward etenson of the fourth-order accurate scheme n one-dmenson defned n Secton 3.4. Wrtten out n detal, the fourth-order accurate upwnd scheme (denoted by UW4-D) that uses Gaussan quadrature n tme wth M =, s gven by where v n+ = v n + t G (k =, k =, k = 3, k 3 = ), 4 () u n+ = u n + tv n + t G (k =, k = 6, k =, k 3 = ), 8 () G (k p ) = k L 4h u n + k t L 4h v n + k t L hu + k 3 6 t3 L hv + 5k 88 ( c h 5 (D + D ) 3 + c y h 5 y(d +y D y ) 3) v k 8 t M hu + k 4 t N hu k t M h v + k 48 t N h v + k 3 t3( c h D + D + c y h y D +y D y ) L hu. () L 4h c D + D ( h D +D ) + c yd +y D y ( h y D +yd y ), M h ( c h 3 (D + D ) + c y h 3 y(d +y D y ) ) L h, ) N h c c y (c y h c h y )(D + D )(D +y D y ) (h D + D h yd +y D y. Note that L 4h s a fourth-order accurate dscretzaton of the operator L.. Stablty regon UW4 D.8 λ y.6.4. Hgh wave number stablty Low wave number stablty Overall stablty Smplfed stablty λ Fgure : Stablty regon for the two-dmensonal fourth-order accurate scheme UW4-D. The black curve denotes the boundary of the stablty regon. The red and blue curves denote the stablty regons for hgh- and low-wave numbers, respectvely. The dashed blue curve denotes the smplfed curve whch s used n practce to determne the tme-step t. Lemma 4.4. The upwnd scheme UW4-D ()-() s fourth-order accurate and stable provded t Λ (d) 4 where Λ (d) 4 =.75 and σ =.75. Note that ths bound s not strct. ( c h ) σ + ( cy h y ) σ ] /σ, (3) 6

27 Proof The proof of accuracy and stablty follows as for the second-order accurate scheme and we are led to the analyss of the two roots z ± = z ± (λ, λ y, ξ, ξ y ). Epandng z ± for small ξ and ξ y yelds z ± = ± W + (W ) ± 3! (W )3 + 4! (W )4 + O((ξ + ξ y ) 5 ), = e ω t + O((ξ + ξ y ) 5 ), where W = λ ξ + λ yξy. Ths shows that the soluton s fourth-order accurate. For stablty, we analyze the magntudes of the roots z ±. The results are summarzed n Fgure. There are four curves shown n ths fgure. The black curve, labeled Overall stablty, s the level contour of the functon Z(λ, λ y ) = ma z ±(λ, λ y, ξ, ξ y ). ξ π, ξ y π To determne ths contour, the functon Z was evaluated numercally by choosng a fne dscretzaton n ξ, ξ y, λ and λ y. The scheme s stable where Z(λ, λ y ). The green curve, labeled Low-wave number stablty, s the stablty curve for ξ and ξ y (an analytc epresson for ths curve can be determned from an analyss of the functon Z for small ξ and ξ y ). Recall that n one-dmenson, the stablty of the scheme s determned for small values of ξ. In two-dmensons these low modes determne the stablty when λ θλ y or λ y θλ where θ.59. The red curve, labeled Hgh-wave number stablty, s the stablty curve for the hgh-frequency modes ξ = ±π and ξ y = ±π (the analytc epresson for ths curve can also be found). The red curve determnes the stablty bound when λ > θλ y and λ y > θλ. The blue curve, labeled Smplfed stablty, was determned by assumng the form λ σ + λ σ y = b σ. and lookng for approprate values for b and σ. The choce b =.75 and σ =.75 was found to gve a curve that s a good appromaton to (but falls wthn) the overall stablty curve. These choces of b and σ lead to the condton (3). 5. Accuracy, stablty, ponts per wavelength and remarks Ths secton summarzes some of the propertes of the upwnd schemes whch have been developed. The stablty bounds, normal mode error formulae and the form of the modfed equatons for the dfferent schemes are presented. In addton, the accuracy requrements of the schemes n terms of ponts per wavelength are dscussed. The secton ends wth a few more general remarks. Fgure lsts the tme-step restrcton n terms of the stablty bounds, c t/h Λ p, for the dfferent schemes n one-dmenson. The fgure also shows the stablty regons for the two-dmensonal schemes wth orders of accuracy one, two and four. Note that the fourth-order scheme has a mamum stable tme step that s appromately twce that of the second-order scheme (along the as λ = λ y n two-dmensons the stablty bounds for the second- and fourth-order schemes are.5 and. respectvely). As a result, we fnd that the ncreased work assocated wth the fourth-order scheme s nearly offset by the larger allowable tme step. Remark. We note that the varaton n the mamum stable tme-step for the varous schemes s qute large. There s a sgnfcant dfference even between schemes UWa and UW, despte the fact that these two schemes are both frst-order accurate and use the same number of ponts n the dscretzaton stencl. For a gven accuracy, upwnd schemes use a wder stencl than would be needed by a compact centered scheme. There s thus more room for varaton n upwnd schemes whch lkely leads to more varaton n the tme-step restrcton. We contrast ths wth the compact hgh-order accurate centered schemes for the second-order wave equaton presented n ] where the mamal stable tme-step s ndependent of the order of accuracy, or the hgh-order accurate schemes for the scalar frst-order wave equaton where the compact odd-order schemes are all stable to a CFL number of one ]. The accuracy of a numercal scheme for wave propagaton problems can be descrbed n terms of the number of ponts per wavelength, N λ, that are requred to propagate a plane wave soluton for a tme of T perods and to a relatve error of ɛ. Ths measure of accuracy was orgnally proposed by Kress and Olger 3] and has subsequently found wde-spread use. The general form of N λ for a p th -order accurate scheme to the frst-order wave equaton s 4] N λ K p ( T/ɛ ) /p, (4) 7

28 Stablty regons for the D upwnd schemes Stablty bounds p scheme Λ p a UWa.89 UW UW.68 4 UW4.9 6 UW6.95 λ y UW D UW D UW4 D (smplfed) UW4 D..4.6 λ.8 Fgure : Left: stablty bounds, c t/h Λ p, for the one-dmensonal upwnd schemes wth orders of accuracy one to sth. Rght: stablty regons for the two-dmensonal upwnd schemes wth orders of accuracy one to four. where the parameter K p depends on the detals of the scheme. Let N (p) λ denote the number of ponts per wavelength requred by the p th -order accurate upwnd schemes for the second-order wave equaton. The form (4) also apples to the second-order wave equaton (as shown below) and K p can be determned for our upwnd schemes. The results are summarzed n Fgure 3 whch not only shows K p for the dfferent schemes but also presents the relatonshps between the coeffcents n the modfed equaton, the accuracy of the normal-mode analyss egenvalues z ± and the number of ponts-per-wavelength, N (p) λ. The results show that the parameter M p (λ) that appears n the leadng error term n the modfed equaton also appears n the error of the egenvalues z ± and the equatons for N (p) λ. To understand the reason for these relatonshps we can look for solutons of the form e (k ωpt) to the modfed equaton () (vald for p =, 4, 6), whch gves the dsperson relaton for the modfed equaton, ω p c k c M p h p (k) p+ + O(h p+ ), (5) whence, ω p ±ck( + M ph p (k) p ) + O(h p+ ). (6) Thus, denotng the eact soluton as ω = ±ck, and notng that ξ = kh and λ = c t/h, t follows that e ωp t e ω t = ±ck t M ph p (k) p + O( t h p ), = ± p+ λ M pξ p+ + O( t h p ). (7) Solutons to the modfed equaton should accurately appromate solutons to the dfference equatons for ξ and thus we epect that z = e ωp t + O( th p+ ). Snce ω t = λξ, equaton (7) thus agrees wth () (for p =, 4, 6). Note that () was ndependently determned from the dscrete appromatons and normal mode theory. The number of ponts per wavelength s defned from the error n computng the plane wave soluton e k(±ct) over T perods. If ɛ s the relatve error at tme t = n t, then from (), ɛ nλ M pξ p. (8) The number of ponts per wavelength N (p) λ s related to ξ by ξ = π/n (p) λ. Snce the perod of the wave s P = π/(ck) and the number of perods T at tme t s T = t/p t follows that ɛ π M p (π) p T N (p) λ, (9) 8

29 whch gves (). In Fgure 4 we plot the parameters K p (λ) versus λ for λ, ]. These results show that the accuracy of the UW scheme s poor for small λ but very good as λ approaches. The accuracy of UW mproves somewhat as λ becomes smaller, whle the accuracy of the UW4 and UW6 schemes do not vary much wth λ. Fgure 4 also shows plots of N (p) λ versus ɛ/t for λ =.5. The results show the dramatc decrease n for the hgh-order accurate schemes, especally when ɛ/t s small (.e. small error tolerances or long tme ntegratons). For eample, wth ɛ/t = 4 the scheme UW would requre appromately 5, ponts per wavelength to acheve the same accuracy as the scheme UW6 usng appromately 5 ponts per wavelength. N (p) λ U t = c U + c M p (λ)h p p+ U p+ + c C p(λ)h p+ p+3 U + O(hp+ t p+ ), p =, 4, 6, () z ± e ±λξ = (±) p+ λ M p(λ) ξ p+ + O(ξ p+ ), p =,, 4, 6, () N (p) ( ) ( /p, /p, λ K p T/ɛ Kp = π π M p (λ) ) p =,, 4, 6. () p scheme M p (λ) UW ( λ) UW ( + 3λ λ ) 4 UW4 6 ( 4 35λ + 6λ + 45λ 3 6λ 4 ) 6 UW6 8 (8 + 47λ 49λ 7λ 3 + λ 4 97λ 5 ) Fgure 3: The relatonshp between the coeffcents M p(λ), the general form of the modfed equatons (), the mode analyss solutons z ± and the number of ponts per wavelength N (p) λ for the upwnd schemes of dfferent orders of accuracy. Note that the modfed equaton for p = s the only one not of the form () Ponts per wavelength, K p verses λ K (UW) K (UW) K 4 (UW4) K (UW6) 6 Ponts per wavelength Ponts per wavelength versus ε/t (λ=.5) UW UW UW4 UW λ 4 3 ε/t Fgure 4: Ponts per wavelength. Left: the ponts per wavelength parameters K p as a functon of the CFL parameter λ. Rght: the ponts per wavelength requrements of the dfferent schemes as a functon of ɛ/t for λ =.5. We conclude ths secton wth some short remarks. Remark : Cache effcency: The space-tme schemes are one-step schemes and can be made very effcent n terms of access to the computer memory and cache. The entre soluton can be updated n a sngle loop, 9

30 thus passng through the data a sngle tme. We have used automatc code generaton to produce optmzed routnes for the schemes from symbolc representatons of the appromatons (usng the Maple symbolc algebra package). Remark : Storage requrements: The storage requred to mplement the upwnd schemes developed heren s essentally equvalent to that requred for the standard C scheme (7), even though the addtonal varable v has been ntroduced. An effcent mplement of the C scheme requres just two tme levels to be stored, u n and u n. However, an effcent mplementaton of the upwnd schemes (even n multple space dmensons) only requres storage for u n and v n, plus some small addtonal lower-dmensonal work space. Therefore the standard and upwnd schemes have asymptotcally the same storage requrements. Remark 3: Method of lnes: Method-of-lnes upwnd schemes usng Runge-Kutta or mult-step methods can be defned n a straght-forward manner from the developments presented here. Remark 4: Confrmaton of the stablty bounds: The analytcally derved stablty bounds of the oneand two-dmensonal schemes were confrmed numercally to wthn one percent. Remark 5: Use of the upwnd flu: Interestngly, the use of the upwnd flu s found to be crtcal to the stablty of all but the second-order accurate scheme. In partcular, f the upwnd flu, formula (3) s replaced by a flu that does not nclude the second term nvolvng the jump n v, then the resultng algorthms, wth the ecepton of the second-order algorthm, wll be uncondtonally unstable for any tme step. The resultng second-order accurate algorthm s neutrally stable wth an amplfcaton factor whose magntude s unformly one. 6. Numercal eamples We now present some numercal results to demonstrate the accuracy and behavor of the upwnd schemes for the second-order wave equaton. We begn n Secton 6. by eamnng the errors and convergence rates of the schemes n one dmenson for a travelng sne wave eact soluton. The convergence rates for the more dffcult top-hat problem, ntroduced prevously, are presented n Secton 6.. The behavor of the schemes for a smooth two-dmensonal surface wave problem wth known soluton s studed n Secton 6.3. Fnally n Secton 6.4 we show results for a dffcult two-dmensonal verson of the top-hat problem. Convergence of the appromatons wll be measured usng the L -norm (ma-norm) and the dscrete L -norm. In general, the dscrete L p -norm of a grd functon u s defned as ] /p u p = u p, N where the sum s taken over all grd ponts and N s the total number of grd ponts. In subsequent dscussons the terms L -norm and L -norm wll be used to ndcate the correspondng dscrete norms. 6.. One-dmensonal travelng sne wave The eact soluton of a travelng sne wave, u(, t) = sn (π( ct)), can be used to demonstrate the accuracy of the upwnd schemes n one dmenson. The problem s solved on the perodc nterval, ] wth c = and ntegrated to a fnal tme t = (one perod). For each scheme, the tme-step s chosen to be.9 tmes the mamum allowable tme-step for that scheme (.e. a CFL number of.9). Fgure 5 shows the results of a convergence study for the frst-order (UW), second-order (UW), fourth-order (UW4), and sth-order (UW6) accurate schemes for ths problem for a sequence of meshes wth ncreasng resoluton. Results are shown for the L -norm (ma-norm) of the error. The four scheme are seen to converge as epected. Gven that the addtonal cost per tme-step to evaluate the hgh-order schemes s not much more than the lower order accurate schemes, these results show the clear beneft of usng hgh-order accurate schemes n terms of the effort requred to reach a requred level of accuracy. The behavor of the hgh-resoluton scheme HR compared to the second-order accurate scheme UW for ths problem s presented n Fgure 6 where both the ma- and L -norms of the error are shown. From the fgure t s clear that the second-order scheme converges as epected, but the convergence character of the hgh-resoluton scheme s somewhat more comple. For the hgh-resoluton scheme, u converges at second order for both L - and L -norms. However, the velocty v converges as O(h ) n the L -norm but the degraded rate of O(h 4/3 ) for the L -norm. Ths somewhat surprsng behavor s the result of the sharp 3

31 SOS Upwnd: Ma Norm errors n u, Sne SOS Upwnd: Ma Norm errors n v, Sne Ma norm u errors UW: e u (p=) UW: e u (p=) UW4: e u (p=4) UW6: e u (p=6) h Ma norm v errors UW: e v (p=) UW: e v (p=) UW4: e v (p=4) UW6: e v (p=6) h Fgure 5: Travelng sne wave: convergence results for the L -norm errors at t =.. Results are presented for the frst-, second-, fourth-, and sth-order accurate schemes. Reference lnes of the correspondng order are dsplayed n black. At left are the results for u and at rght the results for v. SOS Upwnd: Errors n UW and HR, Sne SOS Upwnd: Errors n UW and HR, Sne u errors 3 UW: e u (p=) UW: e u (p=) HR: e u (p=) HR: e u (p=) h v errors 3 UW: e v (p=) UW: e v (p=) HR: e v (p=4/3) HR: e v (p=) h Fgure 6: Travelng sne wave: convergence results for the L -norm and L -norm errors for the UW and HR schemes showng the degradaton n convergence for the velocty n the HR scheme due to the lmter. Left: results for u. Rght: results for v. The L -norm errors for v are seen to converge at a rate p = 4/3. All other cases converge at p =. swtch ntroduced by the MnMod lmter and s smlar to behavor ehbted by hgh-resoluton schemes for the frst-order system 5]. 6.. One-dmensonal top-hat problem The top-hat problem was ntroduced earler n Secton. The ntal condtons are (3)-(4) and the eact soluton s gven by (5)-(6). As noted prevously, ths s a very dffcult problem snce the soluton for u does not have a contnuous frst dervatve, whle v s defned n terms of Drac delta functons. The behavor of the dscrete soluton usng all fve schemes s shown n Fgure 7. Ths s a rather weak soluton and t s somewhat surprsng that a numercal method can provde reasonable appromatons n ths regme. Nonetheless we see that all methods provde qute good results. The trend from low to hgh order s clear from the plots. Also clear s the effect of usng the lmter n the hgh-resoluton scheme where the oscllatons near the dscontnuty are sgnfcantly suppressed n comparson to the unlmted second order scheme. We would lke to draw partcular attenton however, to the hgh qualty of the results even for the hgh order schemes. The sth-order scheme for nstance has surprsngly mld oscllatons near the dscontnuty whle capturng the jump wth only a few ponts. Smlar phenomenology s dscussed n 6, ] for dscretzatons of the frst-order wave equaton. Also note that the sze of the dscrete spke for the Drac delta functon n v s a rather good ndcator of the resolvng power of the varous schemes. Although the L -norm errors are a good measure of the convergence of u, t s not a good measure for 3

32 Second Order Wave Equaton: u, Top Hat t=.5, N= eact UW UW HR UW4 UW6.5. Second Order Wave Equaton: u, Top Hat t=.5, N= eact UW UW HR UW4 UW Second Order Wave Equaton: v, Top Hat t=.5, N= eact UW UW HR UW4 UW6 5 5 Second Order Wave Equaton: v, Top Hat t=.5, N= eact UW UW HR UW4 UW Fgure 7: Top-hat: numercal solutons usng the frst-order (UW), second-order (UW), second-order hghresoluton (HR), fourth-order (UW4), sth-order (UW6) schemes. The solutons for u (top) and v (bottom) are shown at t =.5, computed usng N = grd ponts. the convergence of the delta functons n v. Instead we measure the L -norm error n the ntegral of v, ˆv(, t) = v(ξ, t) dξ whch s n some sense a measure of the sze and locaton of the delta functons n v. Fgure 8 shows the convergence character of the fve schemes for u and ˆv. All the schemes under consderaton converge at the rate O(h p/p+ ) where p s the nomnal convergence rate for the scheme for smooth problems. These are the epected convergence rates for numercal appromatons to the frst-order system wth jump ntal data 5] A two-dmensonal surface wave problem Surface waves are an nterestng phenomena assocated wth wave equatons and we consder the numercal smulaton of such a wave as a verfcaton of our upwnd schemes n two-dmensons. Consder the ntal 3

33 SOS Upwnd: L norm errors n u, Top Hat t=.5 SOS Upwnd: L norm errors n v, Top Hat t=.5 L norm u errors 3 UW: e u (p=/) UW: e u (p=/3) HR: e u (p=/3) UW4: e u (p=4/5) UW6: e u (p=6/7) 3 h Fgure 8: Top-hat: L convergence of u and V = R reference and have slopes for p =,, 4, 6. p p+ L norm v errors 3 UW: e v (p=/) UW: e v (p=/3) HR: e v (p=/3) UW4: e v (p=4/5) UW6: e v (p=6/7) 3 h v(ξ, t) dξ at t =.5. The sold black lnes are drawn for boundary value problem for the second-order wave equaton n two-dmensons, u t = u c + c y u(, y, ) = u (, y), u, ( π, π) (, ), (3) y u t (, y, ) = v (, y), (4) u (,, t) = αu(,, t), u( + π, y, t) = u(, y, t), (5) y where α R. We look for travelng surface wave solutons to these equatons of the form u = e βy e (k ωt) whch are perodc n and decay to zero as y. Solutons of ths type do est and are gven by u(, y, t) = A k e αy cos(k ± ωt + φ k ), k =, ±, ±,..., (6) where A k and φ k are constants and where ω satsfes the dsperson relaton ω = c k c yα. Solutons that reman bounded n tme and decay to zero as y requre < α (c /c y )k and A = (snce the k = mode wth grow eponentally n tme f α ). The phase velocty of the surface waves (6) s gven by c s ω k = c c yα k. (7) The waves are dspersve wth long wave lengths (k small) propagatng more slowly than short wavelengths (k large). Note that mode k becomes statonary f α = c c y k, whle the speed of propagaton of short wavelengths approaches c as k. Equatons (3)-(5) are solved numercally for the dsplacement, u n, and velocty, vn, wth ntal condtons for u and v taken from the eact soluton at t =. At y =, boundary condtons are appled to both dsplacement and velocty. The boundary condton u y (,, t) = αu(,, t) s dfferentated wth respect to tme to yeld v y (,, t) = αv(,, t). For the frst- and second-order accurate schemes, UW, UW, and HR, the boundary condtons are mposed usng a second- and fourth-order accurate appromaton to u y (,, t) = αu(,, t), ( u n h,j+ u n ),j = αu n,j, y (8) ( u n h,j 8u n,j + 8u n,j+ u n ),j+ = αu n,j, (9) where j s the nde correspondng to the cell wth center on the boundary y = and j + and j + denote the ndces of the frst and second ghost cells. Smlar equatons are used for vj n. Equatons (8)-(9) are used to determne the values of u n on two lnes of ghost cells, u,j+ and u,j+. Note that the nteror equaton s solved on the boundary cells (, j). 33

34 For the fourth-order accurate scheme, hgh-order compatblty condtons are derved by repeated dfferentaton of the boundary condton wth respect to tme whch yelds the condtons u yyy (,, t) = αu yy (,, t) and v yyy (,, t) = αv yy (,, t). The values on three ghost lnes for the fourth-order accurate scheme UW4 are determned from a fourth- and sth-order accurate appromaton to u y (,, t) = αu(,, t) together wth a second-order accurate appromaton to u yyy (,, t) = αu yy (,, t), ( u n h,j 8u n,j + 8u n,j+ u n ),j+ = αu n,j, y ( u n h 3,j + u n,j u n,j+ + u n α (,j+) = u n y h,j + 6u n,j 3u n,j + 6u n,j+ u n ),j+, y ( u n 6h,j 3 + 9u n,j 45u n,j + 45u n,j+ 9u n,j+ + u n ),j+3 = αu n,j. (3) y Smlar equatons are used to obtan the values of v n at the ghost ponts. For practcal reasons, the doman s truncated n the y-drecton to the nterval y π, ]. The eact soluton s mposed at y = π so that any possble errors arsng from the artfcal truncaton of the doman are elmnated. To prevent any eponental growth from the k = mode the appromate soluton s projected on the boundary y = to satsfy a dscrete appromaton to π u(,, t) d =. π Ths problem s solved numercally usng the parameters k =, c = /, c y =, α =.4 and A =. For these parameters, the speed of the surface wave s c s =.3. Numercal appromatons are generated usng schemes UW-D, UW-D, HR-D and UW4-D. In each case a CFL number equal to.9 s used, (.e. at 9% of the mamum stable lmt as gven n equaton () for the frst-order scheme, equaton (7) for the second-order scheme, and equaton (3) for the fourth-order scheme). Fgure 9 shows contour plots of the soluton at t = 5 for both u and v. u.3 v.3 Fgure 9: Surface wave solutons of the form (6) for k = at t = 5. At left s dsplacement u and at rght the velocty v. Fgure shows contours of the error n the computed results for the three methods at t = 5 usng 8 grd cells n the and y drectons. A convergence study of the L -norm errors at t = 5 for the dsplacement and velocty s shown n Fgure. The errors n dsplacement and velocty all converge at the epected rates of p for the p th -order accurate schemes. Note that n contrast to the case of the travelng sne wave n Secton 6. when the velocty for the HR scheme converged at a degraded rate of 4/3 n the ma-norm, n ths case the velocty converges at a rate of n the ma-norm. The reason for ths curous result s traced to the fact that the surface wave s travelng at a dfferent speed, c s, than the characterstc wave speed n the -drecton, c. Indeed, the 4/3 rate s reproduced for α = when c s = c. For the case c s = c, the errors generated by the lmter (whch are localzed near mama and mnma n the soluton) move wth the soluton. The errors near the mamum or mnmum thus accumulate over tme, resultng n a degradaton to the mamum norm convergence rates. On the other hand when the speeds c s and c are dfferent, the errors ntroduced by the lmter do not concde wth prevous errors and thus the errors are more spread out; the mamum norm errors are not degraded n ths case A top-hat problem n two space dmensons As a fnal eample consder the soluton to a two-dmensonal verson of the top-hat problem. We solve the two-dmensonal second-order wave equaton (8) on the doman π, π] π, ] and take c = /, 34

35 3 3 e u e u e u 4 e u UW-D UW-D HR-D UW4-D 3 e v e v e v 5 8 e v 8 Fgure : Errors n u and v at t = 5 usng 8 grd cells n the and y drectons. On top are errors n dsplacement whle on bottom are those for velocty. From left to rght the results were computed usng the frst-order scheme, UW-D, the second-order scheme, UW-D, the hgh-resoluton scheme, HR-D, and the fourth-order scheme, UW4-D. Note the changng scales of the color tables for each contour plot. Ma norm errors n u Ma norm errors n v Ma norm errors h UW D (p=) UW D (p=) HR D (p=) UW4 D (p=4) Ma norm norm errors h UW D (p=) UW D (p=) HR D (p=) UW4 D (p=4) Fgure : Surface wave: convergence results for the L -norm errors n u (left) and v (rght) at t = 5. c y =. The ntal condtons are chosen to be { f v(, y, ) =, u(, y, ) = + (y + ) <, otherwse. Perodc boundary condtons are appled n the -drecton, and a zero normal dervatve s appled at the top and bottom boundares. The numercal appromatons to the normal dervatve boundary condtons are gven as n Secton 6.3 wth α =. Fgure shows the computed results usng the fourth-order accurate scheme, UW4-D, at tmes t =, t =.5, and t = 3.. The soluton was computed usng 64 ponts n the - and y-drectons. The supermposed black contour lnes for the mages of u llustrate the remarkably smooth nature of the appromaton 35

36 u t = t =.5 t = 3 v Fgure : Two-dmensonal top-hat: Tme evoluton of u (top) and v (bottom) usng the fourth-order accurate method UW4-D wth 64 ponts n each coordnate drecton. (contour lnes beng a senstve ndcator of oscllatons n the soluton). Note that to present the results more clearly, small regons where u becomes large and negatve have been clpped to blue, whle both large postve and large negatve values of v have been clpped to pnk and blue, respectvely. In order to compare the relatve merts of the varous schemes, Fgure 3 shows contours of the dsplacement for the four schemes, UW-D, UW-D, HR-D and UW4-D. Results are shown from a coarse grd computaton that used cells n each drecton and for a slghtly fner grd computaton that used cells per coordnate drecton. Note that these fgures use the same scale as the very fne computaton n Fgure. In all cases, the schemes are gvng results that are convergng toward the very fne grd soluton. The fgures show the clear beneft of usng the hgh-order accurate upwnd schemes. The basc structure of the soluton can be seen n the results from the frst-order accurate scheme, but the results from the fourthorder accurate scheme show much more detal. There s some evdence of mnor oscllatons n the UW-D and UW4-D schemes near the soluton dscontnutes (the HR-D seems to suppress these oscllatons), but these are consstent wth the one-dmensonal top-hat results. Fgure 4 shows plots of the dsplacement and velocty along the top boundary at y = for the four schemes. These results are from a fne grd computaton whch used 6 cells n each drecton. Ths fgure shows the much enhanced resolvng power of the fourth-order accurate scheme compared to the lower order accurate schemes. Indeed, from a ponts per wavelength argument (see Secton 5) the frst- and second-order accurate schemes would requre on the order of 6 4 and 6 grd ponts respectvely n each drecton to obtan smlar results to the fourth-order accurate scheme. We note that the fne feature near the center of the plot s almost completely lost n results from the frst-order accurate scheme. There s a clear trend n mproved resoluton movng from UW-D, to HR-D, to UW-D and UW4-D. The HR-D scheme does a reasonably good job of representng the dscontnuty near the upper rght (enlarged fgure shown) wth mnor oscllatons. The UW-D and UW4-D schemes show some oscllatons at ths dscontnuty but these are relatvely mld. 36

37 u UW-D UW-D HR-D UW4-D u Fgure 3: Two-dmensonal top-hat: Contours of u at t = 3 for the four schemes; from left to rght: frst-order accurate UW-D, second-order accurate UW-D, hgh-resoluton HR-D, and fourth-order accurate UW4-D. At top are results usng ponts n each coordnate drecton and at bottom are results usng. 7. Conclusons We have presented a methodology to construct upwnd dscretzatons for the wave equaton n secondorder form that avods reformulatng the problem as a frst-order hyperbolc system. To motvate the approach, we demonstrated how a frst-order accurate upwnd scheme could be developed by usng the d Alembert soluton of the second-order wave equaton to eactly advance a pecewse smooth representaton of the soluton. The pecewse smooth soluton at the new tme was then defned by ntegratng the eact soluton over a cell. To generalze the approach we showed how the soluton to a local Remann problem at cell faces can be used to defne the upwnd flu on the face gven appromatons to left and rght based states on the face. Ths localzed form s the key ngredent needed for generalzng the scheme to multple space dmensons, hgh-order accuracy, varable coeffcents and systems of equatons (we leave varable coeffcents and systems to future work). Usng the localzed form of the upwnd flu we then showed how to develop effcent spacetme schemes n one or more space dmensons to arbtrary order of accuracy. A method-of-lnes approach could also be used, although we dd not pursue that approach here. In one space dmenson we developed and analyzed schemes wth orders of accuracy one, two, four and s. A second-order accurate nonlnear hgh-resoluton scheme based on lmters was also developed. Normal mode analyss was used to determne the accuracy of the schemes and the stablty bound on the tme-steps. Modfed equatons were used to elucdate the dsspatve and dspersve nature of the upwnd schemes. The relatonshps between the modfed equatons, the soluton error and the accuracy requrements n terms of ponts per wavelength were also descrbed. In two dmensons we developed and analyzed schemes wth orders of accuracy one, two and four along wth a second-order accurate hgh-resoluton scheme. A seres of numercal tests confrmed the theoretcal results and also demonstrated some of the attractve propertes of the schemes. There are a varety of avenues to consder n future work. Two mportant areas are the etenson of the approach to varable coeffcents and to systems of second-order wave equatons such as Mawell s equatons wrtten n second-order form ], or the elastc wave equaton and nonlnear etensons thereof. To address geometrc complety we wll develop appromatons for curvlnear grds whch can be used n the contet of overlappng grds 7] to develop effcent schemes for comple geometry. As shown n ] for the case of the frst-order system for lnear elastcty, upwnd schemes can be very effectve when solvng wave-propagaton problems on overlappng grds snce they naturally suppress possble nstabltes that can be generated from the overlappng grd nterpolaton. Upwnd schemes for second-order wave equatons may also be useful when 37