A large class of initial value (time-evolution) PDEs in one space dimension can be cast into the form of a flux-conservative equation, u (19.1.

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1 834 Chapter 19. Partial Differetial Equatios egieerig; these methods allow cosiderable freedom i puttig computatioal elemets where you wat them, importat whe dealig with highly irregular geometries. Spectral methods [13-15] are preferred for very regular geometries ad smooth fuctios; they coverge more rapidly tha fiite-differece methods (cf. 19.4), but they do ot work well for problems with discotiuities. CITED REFERENCES AND FURTHER READING: Ames, W.F. 1977, Numerical Methods for Partial Differetial Equatios, 2d ed. (New York: Academic Press). [1] Richtmyer, R.D., ad Morto, K.W. 1967, Differece Methods for Iitial Value Problems, 2d ed. (New York: Wiley-Itersciece). [2] Roache, P.J. 1976, Computatioal Fluid Dyamics (Albuquerque: Hermosa). [3] Mitchell, A.R., ad Griffiths, D.F. 1980, The Fiite Differece Method i Partial Differetial Equatios (New York: Wiley) [icludes discussio of fiite elemet methods]. [4] Dorr, F.W. 1970, SIAM Review, vol. 12, pp [5] Meierik, J.A., ad va der Vorst, H.A. 1977, Mathematics of Computatio, vol. 31, pp [6] va der Vorst, H.A. 1981, Joural of Computatioal Physics, vol. 44, pp [review of sparse iterative methods]. [7] Kershaw, D.S. 1970, Joural of Computatioal Physics, vol. 26, pp [8] Stoe, H.J. 1968, SIAM Joural o Numerical Aalysis, vol. 5, pp [9] Jesshope, C.R. 1979, Computer Physics Commuicatios, vol. 17, pp [10] Strag, G., ad Fix, G. 1973, A Aalysis of the Fiite Elemet Method (Eglewood Cliffs, NJ: Pretice-Hall). [11] Burett, D.S. 1987, Fiite Elemet Aalysis: From Cocepts to Applicatios (Readig, MA: Addiso-Wesley). [12] Gottlieb, D. ad Orszag, S.A. 1977, Numerical Aalysis of Spectral Methods: Theory ad Applicatios (Philadelphia: S.I.A.M.). [13] Cauto, C., Hussaii, M.Y., Quarteroi, A., ad Zag, T.A. 1988, Spectral Methods i Fluid Dyamics (New York: Spriger-Verlag). [14] Boyd, J.P. 1989, Chebyshev ad Fourier Spectral Methods (New York: Spriger-Verlag). [15] 19.1 Flux-Coservative Iitial Value Problems A large class of iitial value (time-evolutio) PDEs i oe space dimesio ca be cast ito the form of a flux-coservative equatio, u t = F(u) (19.1.1) x where u ad F are vectors, ad where (i some cases) F may deped ot oly o u but also o spatial derivatives of u. The vector F is called the coserved flux. For example, the prototypical hyperbolic equatio, the oe-dimesioal wave equatio with costat velocity of propagatio v 2 u t 2 = v2 2 u x 2 (19.1.2)

2 19.1 Flux-Coservative Iitial Value Problems 835 ca be rewritte as a set of two first-order equatios where r t = v s x s t = v r x (19.1.3) r v u x s u (19.1.4) t I this case r ad s become the two compoets of u, ad the flux is give by the liear matrix relatio F(u) = ( ) 0 v u (19.1.5) v 0 (The physicist-reader may recogize equatios (19.1.3) as aalogous to Maxwell s equatios for oe-dimesioal propagatio of electromagetic waves.) We will cosider, i this sectio, a prototypical example of the geeral fluxcoservative equatio (19.1.1), amely the equatio for a scalar u, u t = v u (19.1.6) x with v a costat. As it happes, we already kow aalytically that the geeral solutio of this equatio is a wave propagatig i the positive x-directio, u = f(x vt) (19.1.7) where f is a arbitrary fuctio. However, the umerical strategies that we develop will be equally applicable to the more geeral equatios represeted by (19.1.1). I some cotexts, equatio (19.1.6) iscalled a advective equatio, because the quatity u is trasported by a fluid flow with a velocity v. How do we go about fiite differecig equatio (19.1.6) (or, aalogously, )? The straightforward approach is to choose equally spaced poits alog both the t- adx-axes. Thus deote x = x 0 +, t =t 0 +, =0,1,...,J =0,1,...,N (19.1.8) Let u deote u(t,x ). We have several choices for represetig the time derivative term. The obvious way is to set u t = u+1 u + O() (19.1.9), This is called forward Euler differecig (cf. equatio ). While forward Euler is oly first-order accurate i, it has the advatage that oe is able to calculate

3 836 Chapter 19. Partial Differetial Equatios t or FTCS x or Figure Represetatio of the Forward Time Cetered Space (FTCS) differecig scheme. I this ad subsequet figures, the ope circle is the ew poit at which the solutio is desired; filled circles are kow poits whose fuctio values are used i calculatig the ew poit; the solid lies coect poits that are used to calculate spatial derivatives; the dashed lies coect poits that are used to calculate time derivatives. The FTCS scheme is geerally ustable for hyperbolic problems ad caot usually be used. quatities at timestep +1i terms of oly quatities kow at timestep. Forthe space derivative, we ca use a secod-order represetatio still usig oly quatities kow at timestep : u x = u +1 u 1 + O( 2 ) ( ), 2 The resultig fiite-differece approximatio to equatio (19.1.6) is called the FTCS represetatio (Forward Time Cetered Space), u +1 u ( u +1 u ) 1 = v 2 ( ) which ca easily be rearraged to be a formula for u +1 i terms of the other quatities. The FTCS scheme is illustrated i Figure It s a fie example of a algorithm that is easy to derive, takes little storage, ad executes quickly. Too bad it does t work! (See below.) The FTCS represetatio is a explicit scheme. This meas that u +1 for each ca be calculated explicitly from the quatities that are already kow. Later we shall meet implicit schemes, which require us to solve implicit equatios couplig the u +1 for various. (Explicit ad implicit methods for ordiary differetial equatios were discussed i 16.6.) The FTCS algorithm is also a example of a sigle-level scheme, sice oly values at time level have to be stored to fid values at time level +1. vo Neuma Stability Aalysis Ufortuately, equatio ( ) is of very limited usefuless. It is a ustable method, which ca be used oly (if at all) to study waves for a short fractio of oe oscillatio period. To fid alterative methods with more geeral applicability, we must itroduce the vo Neuma stability aalysis. The vo Neuma aalysis is local: We imagie that the coefficiets of the differece equatios are so slowly varyig as to be cosidered costat i space ad time. I that case, the idepedet solutios, or eigemodes, of the differece equatios are all of the form u = ξ e ik ( )

4 19.1 Flux-Coservative Iitial Value Problems 837 t or Lax x or Figure Represetatio of the Lax differecig scheme, as i the previous figure. The stability criterio for this scheme is the Courat coditio. where k is a real spatial wave umber (which ca have ay value) ad ξ = ξ(k) is a complex umber that depeds o k. The key fact is that the time depedece of a sigle eigemode is othig more tha successive iteger powers of the complex umber ξ. Therefore, the differece equatios are ustable (have expoetially growig modes) if ξ(k) > 1 for some k. The umber ξ is called the amplificatio factor at a give wave umber k. To fid ξ(k), we simply substitute ( ) back ito ( ). Dividig by ξ, we get ξ(k)=1 i v si k ( ) whose modulus is > 1 for all k; so the FTCS scheme is ucoditioally ustable. If the velocity v were a fuctio of t ad x, the we would write v i equatio ( ). I the vo Neuma stability aalysis we would still treat v as a costat, theideabeigthatforvslowly varyig the aalysis is local. I fact, eve i the case of strictly costat v, the vo Neuma aalysis does ot rigorously treat the ed effects at =0ad = N. More geerally, if the equatio s right-had side were oliear i u, thea vo Neuma aalysis would liearize by writig u = u 0 + δu, expadig to liear order i δu. Assumig that the u 0 quatities already satisfy the differece equatio exactly, the aalysis would look for a ustable eigemode of δu. Despite its lack of rigor, the vo Neuma method geerally gives valid aswers ad is much easier to apply tha more careful methods. We accordigly adopt it exclusively. (See, for example, [1] for a discussio of other methods of stability aalysis.) Lax Method The istability i the FTCS method ca be cured by a simple chage due to Lax. Oe replaces the term u i the time derivative term by its average (Figure ): u 1 2 ( u +1 + u 1) ( ) This turs ( ) ito u +1 = 1 ( u u v ( 1) u 2 +1 u 1) ( )

5 838 Chapter 19. Partial Differetial Equatios stable ustable t or (a) x or Figure Courat coditio for stability of a differecig scheme. The solutio of a hyperbolic problem at a poit depeds o iformatio withi some domai of depedecy to the past, show here shaded. The differecig scheme ( ) has its ow domai of depedecy determied by the choice of poits o oe time slice (show as coected solid dots) whose values are used i determiig a ew poit (show coected by dashed lies). A differecig scheme is Courat stable if the differecig domai of depedecy is larger tha that of the PDEs, as i (a), ad ustable if the relatioship is the reverse, as i (b). For more complicated differecig schemes, the domai of depedecy might ot be determied simply by the outermost poits. Substitutig equatio ( ), we fid for the amplificatio factor ξ =cosk i v si k ( ) The stability coditio ξ 2 1 leads to the requiremet v 1 ( ) This is the famous Courat-Friedrichs-Lewy stability criterio, ofte called simply the Courat coditio. Ituitively, the stability coditio ca be uderstood as follows (Figure ): The quatity u +1 i equatio ( ) is computed from iformatio at poits 1 ad +1at time. I other words, x 1 ad x +1 are the boudaries of the spatial regio that is allowed to commuicate iformatio to u +1. Now recall that i the cotiuum wave equatio, iformatio actually propagates with a maximum velocity v. If the poit u +1 is outside of the shaded regio i Figure , the it requires iformatio from poits more distat tha the differecig scheme allows. Lack of that iformatio gives rise to a istability. Therefore, caot be made too large. The surprisig result, that the simple replacemet ( ) stabilizes the FTCS scheme, is our first ecouter with the fact that differecig PDEs is a art as much as a sciece. To see if we ca demystify the art somewhat, let us compare the FTCS ad Lax schemes by rewritig equatio ( ) so that it is i the form of equatio ( ) with a remaider term: u +1 u ( u +1 u ) 1 = v + 1 ( u +1 2u + ) u 1 ( ) 2 2 But this is exactly the FTCS represetatio of the equatio (b) u t = v u x + ()2 2 2 u ( )

6 19.1 Flux-Coservative Iitial Value Problems 839 where 2 = 2 / x 2 i oe dimesio. We have, i effect, added a diffusio term to the equatio, or, if you recall the form of the Navier-Stokes equatio for viscous fluid flow, a dissipative term. The Lax scheme is thus said to have umerical dissipatio, or umerical viscosity. We ca see this also i the amplificatio factor. Uless v is exactly equal to, ξ < 1 ad the amplitude of the wave decreases spuriously. Is t a spurious decrease as bad as a spurious icrease? No. The scales that we hope to studyaccurately are those that ecompass may grid poits, so that they have k 1. (The spatial wave umber k is defied by equatio ) For these scales, the amplificatio factor ca be see to be very close to oe, i both the stable ad ustable schemes. The stable ad ustable schemes are therefore about equally accurate. For the ustable scheme, however, short scales with k 1, which we are ot iterested i, will blow up ad swamp the iterestig part of the solutio. Much better to have a stable scheme i which these short wavelegths die away iocuously. Both the stable ad the ustable schemes are iaccurate for these short wavelegths, but the iaccuracy is of a tolerable character whe the scheme is stable. Whe the idepedet variable u is a vector, the the vo Neuma aalysis is slightly more complicated. For example, we ca cosider equatio (19.1.3), rewritte as The Lax method for this equatio is r +1 s +1 [ ] r = t s x [ ] vs vr = 1 2 (r +1 + v r 1 )+ 2 (s +1 s 1 ) = 1 2 (s +1 + s 1)+ v 2 (r +1 r 1) ( ) ( ) The vo Neuma stability aalysis ow proceeds by assumig that the eigemode is of the followig (vector) form, [ ] [ ] r = ξ e ik r 0 ( ) s Here the vector o the right-had side is a costat (both i space ad i time) eigevector, ad ξ is a complex umber, as before. Substitutig ( ) ito ( ), ad dividig by the power ξ, gives the homogeeous vector equatio (cos k) ξ iv si k i v r0 = 0 ( ) si k (cos k) ξ s 0 0 This admits a solutio oly if the determiat of the matrix o the left vaishes, a coditio easily show to yield the two roots ξ s 0 ξ =cosk±i v si k ( ) The stability coditio is that both roots satisfy ξ 1. This agai turs out to be simply the Courat coditio ( ).

7 840 Chapter 19. Partial Differetial Equatios Other Varieties of Error Thus far we have bee cocered with amplitude error, because of its itimate coectio with the stability or istability of a differecig scheme. Other varieties of error are relevat whe we shift our cocer to accuracy, rather tha stability. Fiite-differece schemes for hyperbolic equatios ca exhibit dispersio, or phase errors. For example, equatio ( ) ca be rewritte as ( ξ = e ik + i 1 v ) si k ( ) A arbitrary iitial wave packet is a superpositio of modes with differet k s. At each timestep the modes get multiplied by differet phase factors ( ), depedig o their value of k. If =/v, the the exact solutio for each mode of a wave packet f(x vt) is obtaied if each mode gets multiplied by exp( ik). For this value of, equatio ( ) shows that the fiite-differece solutio gives the exact aalytic result. However, if v/ is ot exactly 1, the phase relatios of the modes ca become hopelessly garbled ad the wave packet disperses. Note from ( ) that the dispersio becomes large as soo as the wavelegth becomes comparable to the grid spacig. A third type of error is oe associated with oliear hyperbolic equatios ad is therefore sometimes called oliear istability. For example, a piece of the Euler or Navier-Stokes equatios for fluid flow looks like v t v = v +... ( ) x The oliear term i v ca cause a trasfer of eergy i Fourier space from log wavelegths to short wavelegths. This results i a wave profile steepeig util a vertical profile or shock develops. Sice the vo Neuma aalysis suggests that the stability ca deped o k, a scheme that was stable for shallow profiles ca become ustable for steep profiles. This kid of difficulty arises i a differecig scheme where the cascade i Fourier space is halted at the shortest wavelegth represetable o the grid, that is, at k 1/. If eergy simply accumulates i these modes, it evetually swamps the eergy i the log wavelegth modes of iterest. Noliear istability ad shock formatio is thus somewhat cotrolled by umerical viscosity such as that discussed i coectio with equatio ( ) above. I some fluid problems,however, shock formatio is ot merely a aoyace, but a actual physical behavior of the fluid whose detailed study is a goal. The, umerical viscosity aloe may ot be adequate or sufficietly cotrollable. This is a complicated subect which we discuss further i the subsectio o fluid dyamics, below. For wave equatios, propagatio errors (amplitude or phase) are usually most worrisome. For advective equatios, o the other had, trasport errors are usually of greater cocer. I the Lax scheme, equatio ( ), a disturbace i the advected quatity u at mesh poit propagates to mesh poits +1ad 1 at the ext timestep. I reality, however, if the velocity v is positive the oly mesh poit +1 should be affected.

8 19.1 Flux-Coservative Iitial Value Problems 841 v t or x or upwid Figure Represetatio of upwid differecig schemes. The upper scheme is stable whe the advectio costat v is egative, as show; the lower scheme is stable whe the advectio costat v is positive, also as show. The Courat coditio must, of course, also be satisfied. The simplest way to model the trasport properties better is to use upwid differecig (see Figure ): u +1 u u u 1, v = v > 0 u +1 ( ) u, v < 0 Note that this scheme is oly first-order, ot secod-order, accurate i the calculatio of the spatial derivatives. How ca it be better? The aswer is oe that aoys the mathematicias: The goal of umerical simulatios is ot always accuracy i a strictly mathematical sese, but sometimes fidelity to the uderlyig physics i a sese that is looser ad more pragmatic. I such cotexts, some kids of error are much more tolerable tha others. Upwid differecig geerally adds fidelity to problems where the advected variables are liable to udergo sudde chages of state, e.g., as they pass through shocks or other discotiuities. You will have to be guided by the specific ature of your ow problem. For the differecig scheme ( ), the amplificatio factor (for costat v)is ξ =1 v (1 cos k) iv si k ( ) ( ) ξ 2 =1 2 v 1 v (1 cos k) ( ) v So the stability criterio ξ 2 1 is (agai) simply the Courat coditio ( ). There are various ways of improvig the accuracy of first-order upwid differecig. I the cotiuum equatio, material origially a distace v away

9 842 Chapter 19. Partial Differetial Equatios t or x or staggered leapfrog Figure Represetatio of the staggered leapfrog differecig scheme. Note that iformatio from two previous time slices is used i obtaiig the desired poit. This scheme is secod-order accurate i both space ad time. arrives at a give poit after a time iterval. I the first-order method, the material always arrives from away. If v (to isure accuracy), this ca cause a large error. Oe way of reducig this error is to iterpolate u betwee 1 ad before trasportig it. This gives effectively a secod-order method. Various schemes for secod-order upwid differecig are discussed ad compared i [2-3]. Secod-Order Accuracy i Time Whe usig a method that is first-order accurate i time but secod-order accurate i space, oe geerally has to take v sigificatly smaller tha to achieve desired accuracy, say, by at least a factor of 5. Thus the Courat coditio is ot actually the limitig factor with such schemes i practice. However, there are schemes that are secod-order accurate i both space ad time, ad these ca ofte be pushed right to their stability limit, with correspodigly smaller computatio times. For example, the staggered leapfrog method for the coservatio equatio (19.1.1) is defied as follows (Figure ): Usig the values of u at time t, compute the fluxes F. The compute ew values u+1 usig the time-cetered values of the fluxes: u +1 u 1 = (F +1 F 1 ) ( ) The ame comes from the fact that the time levels i the time derivative term leapfrog over the time levels i the space derivative term. The method requires that u 1 ad u be stored to compute u +1. For our simple model equatio (19.1.6), staggered leapfrog takes the form u +1 u 1 = v (u +1 u 1) ( ) The vo Neuma stability aalysis ow gives a quadratic equatio for ξ, rather tha a liear oe, because of the occurrece of three cosecutive powers of ξ whe the

10 19.1 Flux-Coservative Iitial Value Problems 843 form ( ) for a eigemode is substituted ito equatio ( ), ξ 2 1= 2iξ v si k ( ) whose solutio is ξ = i v si k ± 1 ( ) 2 v si k ( ) Thus the Courat coditio is agai required for stability. I fact, i equatio ( ), ξ 2 =1for ay v. This is the great advatage of the staggered leapfrog method: There is o amplitude dissipatio. Staggered leapfrog differecig of equatios like ( ) is most trasparet if the variables are cetered o appropriate half-mesh poits: r+1/2 v u x s +1/2 u t +1/2 +1/2 = v u +1 u = u+1 u ( ) This is purely a otatioal coveiece: we ca thik of the mesh o which r ad s are defied as beig twice as fie as the mesh o which the origial variable u is defied. The leapfrog differecig of equatio ( ) is r +1 +1/2 r +1/2 s +1/2 s 1/2 = s+1/2 +1 s +1/2 = v r +1/2 r 1/2 ( ) If you substitute equatio ( ) i equatio ( ), you will fid that oce agai the Courat coditio is required for stability, ad that there is o amplitude dissipatio whe it is satisfied. If we substitute equatio ( ) i equatio ( ), we fid that equatio ( ) is equivalet to u +1 2u + u 1 () 2 = v 2 u +1 2u + u 1 () 2 ( ) This is ust the usual secod-order differecig of the wave equatio (19.1.2). We see that it is a two-level scheme, requirig both u ad u 1 to obtai u +1. I equatio ( ) this shows up as both s 1/2 ad r beig eeded to advace the solutio. For equatios more complicated tha our simple model equatio, especially oliear equatios, the leapfrog method usually becomes ustable whe the gradiets get large. The istability is related to the fact that odd ad eve mesh poits are completely decoupled, like the black ad white squares of a chess board, as show

11 844 Chapter 19. Partial Differetial Equatios Figure Origi of mesh-drift istabilities i a staggered leapfrog scheme. If the mesh poits are imagied to lie i the squares of a chess board, the white squares couple to themselves, black to themselves, but there is o couplig betwee white ad black. The fix is to itroduce a small diffusive mesh-couplig piece. i Figure This mesh driftig istability is cured by couplig the two meshes through a umerical viscosity term, e.g., addig to the right side of ( ) a small coefficiet ( 1) times u +1 2u + u 1. For more o stabilizig differece schemes by addig umerical dissipatio, see, e.g., [4]. The Two-Step Lax-Wedroff scheme is a secod-order i time method that avoids large umerical dissipatio ad mesh driftig. Oe defies itermediate values u +1/2 at the half timesteps t +1/2 ad the half mesh poits x +1/2.These are calculated by the Lax scheme: u +1/2 +1/2 = 1 2 (u +1 + u ) 2 (F +1 F ) ( ) Usig these variables, oe calculates the fluxes F +1/2 +1/2. The the updated values are calculated by the properly cetered expressio u +1 u +1 = u ( ) F +1/2 +1/2 F +1/2 1/2 ( ) The provisioal values u +1/2 +1/2 are ow discarded. (See Figure ) Let us ivestigate the stability of this method for our model advective equatio, where F = vu. Substitute ( ) i ( ) to get u +1 [ 1 = u α 2 (u +1 + u ) 1 2 α(u +1 u ) 1 ] ( ) 2 (u + u 1 )+1 2 α(u u 1 )

12 19.1 Flux-Coservative Iitial Value Problems 845 two-step Lax Wedroff halfstep poits t or x or Figure Represetatio of the two-step Lax-Wedroff differecig scheme. Two halfstep poits ( ) are calculated by the Lax method. These, plus oe of the origial poits, produce the ew poit via staggered leapfrog. Halfstep poits are used oly temporarily ad do ot require storage allocatio o the grid. This scheme is secod-order accurate i both space ad time. where The so α v ( ) ξ =1 iα si k α 2 (1 cos k) ( ) ξ 2 =1 α 2 (1 α 2 )(1 cos k) 2 ( ) The stability criterio ξ 2 1 is therefore α 2 1, or v as usual. Icidetally, you should ot thik that the Courat coditio is the oly stability requiremet that ever turs up i PDEs. It keeps doig so i our model examples ust because those examples are so simple i form. The method of aalysis is, however, geeral. Except whe α =1, ξ 2 <1i ( ), so some amplitude dampig does occur. The effect is relatively small, however, for wavelegths large compared with the mesh size. If we expad ( ) for small k, wefid ξ 2 =1 α 2 (1 α 2 ) (k) ( ) The departure from uity occurs oly at fourth order i k. This should be cotrasted with equatio ( ) for the Lax method, which shows that ξ 2 =1 (1 α 2 )(k) ( ) for small k.

13 846 Chapter 19. Partial Differetial Equatios I summary, our recommedatio for iitial value problems that ca be cast i flux-coservative form, ad especially problems related to the wave equatio, is to use the staggered leapfrog method whe possible. We have persoally had better success with it tha with the Two-Step Lax-Wedroff method. For problems sesitive to trasport errors, upwid differecig or oe of its refiemets should be cosidered. Fluid Dyamics with Shocks As we alluded to earlier, the treatmet of fluid dyamics problems with shocks has become a very complicated ad very sophisticated subect. All we ca attempt to do here is to guide you to some startig poits i the literature. There are basically three importat geeral methods for hadlig shocks. The oldest ad simplest method, iveted by vo Neuma ad Richtmyer, is to add artificial viscosity to the equatios, modelig the way Nature uses real viscosity to smooth discotiuities. A good startig poit for tryig out this method is the differecig scheme i 12.11of [1]. This scheme is excellet for early all problems i oe spatial dimesio. The secod method combies a high-order differecig scheme that is accurate for smooth flows with a low order scheme that is very dissipative ad ca smooth the shocks. Typically, various upwid differecig schemes are combied usig weights chose to zero the low order scheme uless steep gradiets are preset, ad also chose to eforce various mootoicity costraits that prevet ophysical oscillatios from appearig i the umerical solutio. Refereces [2-3,5] are a good place to start with these methods. The third, ad potetially most powerful method, is Goduov s approach. Here oe gives up the simple liearizatio iheret i fiite differecig based o Taylor series ad icludes the oliearity of the equatios explicitly. There is a aalytic solutio for the evolutio of two uiform states of a fluid separated by a discotiuity, the Riema shock problem. Goduov s idea was to approximate the fluid by a large umber of cells of uiform states, ad piece them together usig the Riema solutio. There have bee may geeralizatios of Goduov s approach, of which the most powerful is probably the PPM method [6]. Readable reviews of all these methods, discussig the difficulties arisig whe oe-dimesioal methods are geeralized to multidimesios, are give i [7-9]. CITED REFERENCES AND FURTHER READING: Ames, W.F. 1977, Numerical Methods for Partial Differetial Equatios, 2d ed. (New York: Academic Press), Chapter 4. Richtmyer, R.D., ad Morto, K.W. 1967, Differece Methods for Iitial Value Problems, 2d ed. (New York: Wiley-Itersciece). [1] Cetrella, J., ad Wilso, J.R. 1984, Astrophysical Joural Supplemet, vol. 54, pp , Appedix B. [2] Hawley, J.F., Smarr, L.L., ad Wilso, J.R. 1984, Astrophysical Joural Supplemet, vol. 55, pp , 2c. [3] Kreiss, H.-O. 1978, Numerical Methods for Solvig Time-Depedet Problems for Partial Differetial Equatios (Motreal: Uiversity of Motreal Press), pp. 66ff. [4] Harte, A., Lax, P.D., ad Va Leer, B. 1983, SIAM Review, vol. 25, pp [5] Woodward, P., ad Colella, P. 1984, Joural of Computatioal Physics, vol. 54, pp [6]

14 19.2 Diffusive Iitial Value Problems 847 Roache, P.J. 1976, Computatioal Fluid Dyamics (Albuquerque: Hermosa). [7] Woodward, P., ad Colella, P. 1984, Joural of Computatioal Physics, vol. 54, pp [8] Rizzi, A., ad Egquist, B. 1987, Joural of Computatioal Physics, vol. 72, pp [9] 19.2 Diffusive Iitial Value Problems Recall the model parabolic equatio, the diffusio equatio i oe space dimesio, u t = ( D u ) (19.2.1) x x where D is the diffusio coefficiet. Actually, this equatio is a flux-coservative equatio of the form cosidered i the previous sectio, with F = D u (19.2.2) x the flux i the x-directio. We will assume D 0, otherwise equatio (19.2.1) has physically ustable solutios: A small disturbace evolves to become more ad more cocetrated istead of dispersig. (Do t make the mistake of tryig to fid a stable differecig scheme for a problem whose uderlyig PDEs are themselves ustable!) Eve though (19.2.1) is of the form already cosidered, it is useful to cosider it as a model i its ow right. The particular form of flux (19.2.2), ad its direct geeralizatios, occur quite frequetly i practice. Moreover, we have already see that umerical viscosity ad artificial viscosity ca itroduce diffusive pieces like the right-had side of (19.2.1) i may other situatios. Cosider first the case whe D is a costat. The the equatio u t = u D 2 x 2 (19.2.3) ca be differeced i the obvious way: u +1 u [ u +1 2u = D + ] u 1 () 2 (19.2.4) This is the FTCS scheme agai, except that it is a secod derivative that has bee differeced o the right-had side. But this makes a world of differece! The FTCS scheme was ustable for the hyperbolic equatio; however, a quick calculatio shows that the amplificatio factor for equatio (19.2.4) is ξ =1 4D () 2 si2 ( ) k 2 (19.2.5) The requiremet ξ 1leads to the stability criterio 2D () 2 1 (19.2.6)