Numerical methods for hyperbolic balance laws with discontinuous flux functions and applications in radiotherapy

Transcription

1 Numercal methods for hyperbolc balance laws wth dscontnuous flux functons and applcatons n radotherapy Numersche Verfahren für hyperbolsche Blanzglechungen mt unstetger Flussfunkton und Anwendungen n der Strahlentherape von Nadne Pawltta Masterarbet n Mathematk vorgelegt der Fakultät für Mathematk, Informatk und Naturwssenschaften der Rhensch-Westfälschen Technschen Hochschule Aachen Angefertgt be Prof. Dr. Martn Frank Lehrstuhl für Mathematk CCES Zwetgutachter Prof. Dr. Mchael Herty Lehrstuhl C für Mathematk

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3 3 Danksagung Ich möchte mch an deser Stelle be all denen bedanken, de mch be der Anfertgung mener Masterarbet unterstützt haben. Besonderer Dank glt Herrn Professor Martn Frank für de Betreuung deser Arbet und für de velen hlfrechen Anregungen. Zudem bedanke ch mch be den Mtarbetern des MathCCES und menen Freunden und Kommltonen für de freundlche Atmosphäre und fachlche Unterstützung. Ncht zuletzt möchte ch mch auch be mener Famle bedanken, ohne de deses Studum nemals möglch gewesen wäre.

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5 Contents Introducton to hyperbolc conservaton laws 9. Defnton of conservaton laws The advecton equaton Theory of numercal methods Convergence Numercal methods for lnear equatons The Upwnd method Godunov s method The Lax-Wendroff method The advecton equaton wth non constant velocty Theoretcal analyss Methods The Upwnd method on a nonunform grd The Upwnd method on a unform grd The Lax-Wendroff method Comparson and evaluaton of the derved methods Nonlnear partal dfferental equatons Analyss of nonlnear equatons Burgers equaton Numercal methods for nonlnear equatons The Lax-Fredrchs method Nonlnear equatons wth spatally varyng, dscontnuous flux functon The Lax-Fredrchs method on a nonunform grd The Lax-Fredrchs method on a unform grd Flux functons wth arbtrary, pecewse constant coeffcent Applcatons n radotherapy A model for dose calculaton

6 6 Contents 4.2 Soluton of the D model Numercal experments Homogeneous test case One-dmensonal dose calculaton Conclusons and outlook 9

7 Introducton Radotherapy s a way of treatng dseases lke cancer by usng onzng radaton that destroys the cells n the treated area. At the same tme, the damage to the surroundng tssue should be as small as possble. For ths reason, the needed dose of radaton has to be determned accurately. The dffculty that gave rse to ths thess s that the radaton has to pass tssues of dfferent densty. For example, a partcle that travels through ar looses much less energy than one that travels through bone. The densty can be descrbed by a functon ρ that s space dependent and can obtan values from approxmately, whch corresponds to water, to 3, whch s ar. In ths stuaton, the change between the dfferent areas and therefore denstes s abrupt, so ρ s a pecewse constant functon. It can be ascertaned wth the data of a Computer Tomography scan whch s performed before the treatment. The dose of radaton can then be computed by analyzng the transport of the radaton partcles. Ths transport can be descrbed by a hyperbolc balance law, whch s a system of partal dfferental equatons wth a source term. In ths thess, a system of two frst order equatons of the form ɛ Ψ x, ɛ) ρx) x Ψ x, ɛ) =, Ψ x, ɛ) χ ɛ Ψ x, ɛ) ρx) x Ψ x, ɛ) Ψ x, ɛ) s consdered as a model for the radatve transport. )) = T ɛ) Ψ x, ɛ) Usually, there exst several dfferent numercal methods to solve such equatons wth hgh accuracy, but the dfference n the velocty makes them computatonally too expensve and therefore nfeasble. The reason for ths s the so-called CFL condton whch s a necessary condton for stablty of the numercal methods. It establshes a connecton between the tme step sze and the spatal step sze that depends on the densty ρ. For small values of ρ, as s the case when modelng ar, the CFL condton demands a much smaller tme step sze than spatal step sze. Therefore, the exstng standard methods would solve the equaton wth a tme step sze that s only needed for partcles that travel through ar and s much smaller than necessary for partcles that travel through 7

8 8 Introducton water. The am of ths thess s to fnd a numercal method that s better suted for ths problem by adjustng exstng methods so that they no longer requre such a small tme step sze. Note that there does not exst suffcent theoretcal knowledge of such stuatons. Whle t wll be possble to derve methods that solve the gven equatons, there s no theory that provdes the means to evaluate the methods analytcally. In the followng, numercal methods are derved for one-dmensonal systems of equatons by gradually extendng methods for more smple problems. The frst chapter gves a general ntroducton to frst-order, hyperbolc partal dfferental equatons wthout such a dscontnuous coeffcent and ntroduces basc numercal methods for the specal case of lnear equatons. In the second chapter, these methods are modfed to solve lnear equatons wth the addton of a pecewse constant coeffcent whch wll later on be the densty ρ. The results of these two chapters wll be used n the thrd to solve nonlnear equatons, frst wthout and then wth the dscontnuous coeffcent. Ths yelds the bass for the fourth chapter. There, the numercal methods are valdated and appled to dose calculaton n radotherapy.

9 Introducton to hyperbolc conservaton laws Ths chapter ntroduces the bascs of hyperbolc partal dfferental equatons and ther solutons and s mostly based on [] and [2]. After establshng some theoretcal facts about such equatons, the fnte volume method for solvng them s derved and analyzed for convergence to the true soluton. Followng ths, two basc methods for the specal case of a lnear equaton are presented.. Defnton of conservaton laws Consder the ntal value problem or Cauchy problem where f C R) and u L R). ux, t) + fux, t)) =, x R, t >.) t x ux, ) = u x), x R The equaton u t + fu)) x = s the dfferental form of a conservaton law. In order to derve t, examne the one-dmensonal example of the flow of a substance wth densty u through a ppe. The mass between two ponts x and x 2 at tme t can be descrbed by x2 x ux, t)dx. Conservaton of mass then means that the mass of the substance does not change wthn the ppe but rather through flux at the boundares of the ppe. In case ths flux can be descrbed by the functon f, t gves the ntegral form of the conservaton law t x2 Assumng that u and f are smooth, ths results n t x ux, t)dx = f ux, t)) f ux 2, t))..2) x2 x ux, t)dx = x2 x f ux, t)) dx x 9

10 Introducton to hyperbolc conservaton laws whch can be wrtten as x2 x ) ux, t) + f ux, t)) dx =. t x Ths yelds the frst ntroduced dfferental form of the conservaton law... The advecton equaton The advecton equaton ux, t) + a ux, t) =.3) t x s a specal case of.) wth a = f u) snce fu)) x = f u)u x = au x. It models the advecton of a substance n a flud wth velocty a. It s easy to verfy that, gven an ntal condton ux, ) = u x), the soluton of ths equaton s gven by ux, t) = u x at) as t u x at) + a x u x at) = au x at) + au x at) =. Therefore, for arbtrary x, the soluton ux, t) s constant along the so-called characterstcs Xt) = x + at of the equaton because d dt uxt), t) = u txt), t) + X t)u x Xt), t) = u t + au x =. t x Fgure.: Characterstcs for the advecton equaton. Fgure. shows for example some characterstcs for an advecton equaton wth postve

11 Introducton to hyperbolc conservaton laws velocty. The value of the soluton u s the same on each pont of one characterstc, ths means that n practce, the ntal value u smply shfts to the rght n tme, as can be seen n fgure.2. ntal condton, tme t= tme t= tme t= tme t= Fgure.2: Soluton of the advecton equaton at dfferent tme steps. In hgher dmensons, a lnear, constant-coeffcent equaton lke the advecton equaton becomes a lnear system u t + Au x = snce the system u t + fu)) x = can be wrtten as u t + f u)u x = f f s smooth and f u) = A R m m s the Jacoban of f. Such a system s called hyperbolc f the matrx A s dagonalzable wth real egenvalues λ,..., λ m, that s A = R dagλ,..., λ m )R

12 2 Introducton to hyperbolc conservaton laws where R s the matrx that s composed of the correspondng egenvectors. As a result u t + Au x = R u t + R ARR u x = v t + dagλ,..., λ m )v x = v p t + λp vx p =, for p =... m wth R u =: v and therefore, the orgnal system s equvalent to a system of m ndependent advecton equatons. Note that a scalar, frst-order dfferental equaton s always hyperbolc..2 Theory of numercal methods The basc dea of the so-called fnte volume methods s to dvde the spatal doman nto a mesh and to approxmate the ntegral of u over each cell n the mesh. In the one-dmensonal case, one of those cells s an nterval C = x /2, x +/2 ) wth sze x = x +/2 x /2 and x = x n the case of a unform grd as pctured n fgure.3. x t n+ t t n x - x x + Fgure.3: Example for a unform grd. The approxmaton u n be expressed as to the average value of the soluton over C at tme t n can then u n x C ux, t n )dx.

13 Introducton to hyperbolc conservaton laws 3 Ths can be used to construct a numercal method for the soluton of the PDE. The startng pont s the ntegral form.2) of the conservaton law ux, t)dx = f ux t /2, t) ) f ux +/2, t) ). C Integraton over tme and rearrangement gves tn+ ux, t n+ )dx ux, t n )dx = f ux /2, t) ) dt C C t n ux, t n+ )dx = ux, t n )dx x C x C t tn+ f ux x t +/2, t) ) dt t t n tn+ t n tn+ t n f ux +/2, t) ) dt f ux /2, t) ) ) dt where t = t n+ t n s one tme step. The method should therefore have the form wth u n u n+ = u n t x F n +/2 F n /2 ).4) approxmated as before and the average flux F n /2 as F /2 n tn+ f ux t /2, t) ) dt. t n Ths can be computed by means of a numercal flux functon F where t s assumed for now that the flux F n /2 depends only on the values un and un so that F n /2 = Fun, u n )..5) Thus, the method s defned by the flux and approxmates the value u n+ u n+ = u n t Fu n x, u n +) Fu n, u n ) ),.6) whch s the standard formulaton of a fnte volume method. More generally, f the flux F /2 n depends on un m up to un +m, the method s gven by u n+ = u n t Fu n x m+,..., u n +m) Fu n m,..., u n +m ) ). Let N then descrbe the numercal method as N u n m,..., u n +m) = u n t Fu n x m+,..., u n +m) Fu n m,..., u n +m ) ). A method that can be wrtten thusly s called conservatve or n conservaton form. The followng sectons examne ths condton for t before specfyng a method by choosng a numercal flux functon. by

14 4 Introducton to hyperbolc conservaton laws.2. Convergence An obvous property of the numercal method should be that the computed soluton converges to the exact one f the grd s refned, that s for x, t. The Equvalence Theorem of Lax and Rchtmyer see [9]) states that stablty s necessary and suffcent for convergence of a lnear, consstent method. Therefore, there are two condtons that have to be fulflled to acheve convergence. Consstency deals wth the error n one tme step, t s gven when the dfferental equaton s locally well approxmated. For the numercal flux functon F ths means that n the specal case where u s constant and F /2 n satsfy Fu, u) = fu). = fu), F must Stablty s the property that the error of a tme step does not ncrease too much. In the followng, both attrbutes wll be examned n more detal. Consstency Consder the error caused by the applcaton of the numercal method after one tme step. Ths error s called the local truncaton error, t compares the true soluton at a tme t + t wth the true soluton at tme t after one step of the method and s gven as Lx, t) = N ux m x, t),..., ux + m x, t)) ux, t + t))..7) t The numercal method s now sad to be consstent f Furthermore, f the error can be wrtten as lm L., t) =. t L., t) c t p.8) for a constant c and arbtrary, smooth ntal data, the method s of order p. In ths thess, the dscrete -norm x = x s used, though other norms can be used as well, but they mght gve dfferent results. Instead of determnng the order analytcally, t can be estmated graphcally as well. If the rato of x and t s set to be constant, L., t) c t p L., t) c x p

15 Introducton to hyperbolc conservaton laws 5 for some other constant c. Snce t wll later on be mportant to observe the error at every tme step, consder n ths thess from now on the error ɛ := T T xu., t t) N t u.)) t= where N t means that N has been appled t tmes. Therefore, the error ɛ s the weghted sum of the dfference of the exact soluton and the computed one at each pont n space and tme. If the method s now of order p, then ɛ can be estmated by x p as well. For a constant C t then holds that ɛ = C x p log ɛ = log C + p log x. For ths reason, the logarthm of the error can be nterpreted as a straght lne wth slope p. To assess the order p smply compute the error for dfferent values of x and plot them as a functon of log x. Stablty A necessary condton for stablty s the so-called CFL condton, named after Courant, Fredrchs and Lewy [4]. For the advecton equaton.3) t was shown above that the soluton s constant along the characterstcs and ux, t) = u x at) for arbtrary x and t. In other words, the soluton at a pont x, t) depends on the ntal data only at one pont x at. Because of ths, the numercal soluton should as well depend on at least ths pont, or rather on the pont on the respectve characterstc that was computed one tme step before. Ths set of ponts s called the doman of dependence Dx, t) of the PDE at the pont x, t) and smply takes the form Dx, t) = {x at} for the advecton equaton. The CFL condton now states for the general case that a numercal method can only be convergent f the doman of dependence of the PDE s a subset of the numercal doman of dependence. For a method of the form.6), whch computes u n+ usng a three pont stencl consstng of u n,un and u n +, the doman of dependence s the set { Dx, t) = x x x } x t x x + t t t as can be seen n fgure.4. There the value at the pont x, t n+ ) depends on x, t n ), x, t n ) and x +, t n ) and these n turn on x 2, t n ) up to x +2, t n ) and so on, so that the edges of the resultng trangle can be descrbed by t x x x t t and t x+ t t. Hence, for the CFL condton to be fulflled, the doman of dependence of the gven PDE has to be stuated nsde ths trangle.

16 6 Introducton to hyperbolc conservaton laws t n+ t n t n- x -2 x - x x + x +2 Fgure.4: Three pont stencl. For the advecton equaton, the doman of dependence s the set Dx, t) = {x at} and therefore whch can be reduced to x x x t x at x + t t t, a t x where the number ν = a t x s called the courant number. Monotoncty and Total Varaton.9) There are other propertes that are connected to the convergence of a method and wll be analyzed further here. The frst s the concept of monotoncty. It s reasonable to demand that f the soluton s monotone on an nterval, the numercal soluton should be monotone as well. Based on ths noton, a numercal method N s called monotone f and t s called monotoncty-preservng f u n u n j N u n ) N u n j ) u u j u n u n j. Another requrement deals wth the total varaton of a functon. functon g the total varaton s defned as T V g) = lm sup ɛ ɛ gx) gx ɛ) dx. For an arbtrary

17 Introducton to hyperbolc conservaton laws 7 For the specal case of the pecewse constant functon or rather grd functon u n ths smplfes to T V u n ) = u n u n. = The total varaton can be used to measure oscllatons n a functon. To avod the appearance of oscllatons whch are only the result of the computatons and have no physcal meanng, the total varaton of the soluton should not ncrease. A numercal method s therefore called total varaton dmnshng TVD) or more approprately total varaton non-ncreasng f T V u n+ ) T V u n ) for u n+ = N u n ). Furthermore, monotoncty mples TVD and TVD mples monotoncty preservng, so t suffces to proof monotoncty when testng a numercal method for all three propertes..3 Numercal methods for lnear equatons Derve now the basc numercal methods for the soluton of the advecton equaton and analyze them for convergence as descrbed above..3. The Upwnd method It s reasonable to choose the flux functon F /2 n accordng to the structure of the soluton n order to acheve better results. For the constant-coeffcent advecton equaton u t + au x = t was shown that the soluton s constant along the characterstcs, that s ux, t n ) = ux + a t, t n+ ). Hence, f a s postve, ux, t n+ ) depends only on the values of u at tme t n to the left of x and the same should hold for u n+. In other words, the flux F /2 n should be determned by un. Choosng F n /2 = Fun, u n ) = au n n.4) yelds the frst-order upwnd method for the advecton equaton as u n+ = u n a t x un u n )..) It s mportant to observe that ths ansatz s only correct f ux, t n+ ) depends solely on ux, t n ), whch makes the above choce of the flux sensble. Ths s gven by the

18 8 Introducton to hyperbolc conservaton laws CFL condton for the advecton equaton.9) because ux, t n+ ) = ux a t, t n ) and therefore x a t x x = x. Fgure.5 frst depcts the case where the CFL condton s fulflled because t takes more than one tme step for the characterstcs to cross one spatal nterval C. In the second part ths s no longer the case and the CFL condton s breached. t n+ t n+ t n t n x - x x + x - x x + Fgure.5: Two examples where the CFL condton s fulflled and breached. It should be noted that the upwnd method for the advecton equaton s exact f ν = a t x = because then u n+ = u n u n u n ) = u n, whch comples wth ux, t n+ ) = ux a t, t n ) = ux x, t n ) = ux, t n ). The latter s a property of the exact soluton of the PDE and s gven by followng the characterstcs. In most cases though, ν s strctly smaller than, whch leads to the stuaton n fgure.6. Another possblty to derve the upwnd method s to take the convex combnaton of ux, t n ) and ux, t n ) to compute ux, t n+ ) snce ux, t n+ ) = ux a t, t n ) = ux ν x, t n ). Ths s the result of the evaluaton of the lnear nterpolaton polynomal through x and x at the pont x a t. The polynomal has the form whch smplfes to when nsertng x = x a t. u n + un un x x ) = u n un un x x ), x x x u n + νu n u n ) = u n νu n u n )

19 Introducton to hyperbolc conservaton laws 9 t n+ t n x - x -aδt x Fgure.6: The upwnd method for ν <. Propertes of the upwnd method examnng the local truncaton error.7) that s gven as The method s of frst order as can be shown by Lx, t n ) = t ux, t n ) νux, t n ) ux, t n )) ux, t n+ )). Replace ux, t n ) and ux, t n+ ) by ther Taylor seres expanson ux, t n ) = ux, t n ) x u x x, t n ) + 2 x2 u xx x, t n ) + O x 3 ), ux, t n+ ) = ux, t n ) + t u t x, t n ) + 2 t2 u tt x, t n ) + O t 3 ), whch yelds Lx, t n ) = [ ux, t n ) ν ux, t n ) ux, t n ) + x u x x, t n ) ) t 2 x2 u xx x, t n ) O x 3 ) ux, t n ) t u t x, t n ) ] 2 t2 u tt x, t n ) O t 3 ) = [ ν x u x x, t n ) + t 2 ν x2 u xx x, t n ) + νo x 3 ) t u t x, t n ) ] 2 t2 u tt x, t n ) O t 3 ) = a u x x, t n ) + 2 a xu xxx, t n ) + a x O x3 ) u t x, t n ) 2 tu ttx, t n ) t O t3 ). In a last step, use that for the advecton equaton u t + au x = u t = au x u tt = au x ) t = au t ) x = a 2 u xx,

20 2 Introducton to hyperbolc conservaton laws so that Lx, t n ) = 2 a xu xxx, t n ) 2 a2 tu xx x, t n ) + O x 2, t 2 ) = 2 a x a t)u xxx, t n ) + O x 2, t 2 ). Snce the u xx term has the most nfluence on the local truncaton error, t was proven that the method s of frst order n tme and space. Ths can be verfed graphcally as explaned n secton.2.. example for the soluton of the advecton equaton wth velocty a =. Fgure.7 shows an The ntal condton s a part of the arc tangent on the nterval [, ]. Snce the upwnd method then requres a boundary condton for x =, assume for smplcty that the value u n s gven. Furthermore, the Courant number s ν =.9 and the step sze x s gven as /2 wth = 2,..., 2. The plot shows the base logarthm of the error dependng on the logarthm /2 of the step sze. As can be seen, the slope and wth t the order s ndeed. Upwnd least squares approxmaton slope.52) error Fgure.7: The logarthmc error plot of the upwnd method for the soluton of the advecton equaton wth an arc tangent as ntal condton It s easy to verfy further that ths method s TVD and monotoncty preservng as an

21 Introducton to hyperbolc conservaton laws 2 addtonal property snce T V u n+ ) = =.3.2 Godunov s method u n+ u n+ = = = ν)u n u n ) + νu n u n 2) ν) u n u n + = ν)t V u n ) + νt V u n ) = T V u n ). = ν u n u n 2 The upwnd method can be nterpreted as a specal case of Godunov s method whch determnes the values u n+ wth the REA algorhm. Here, REA stands for Reconstructon, Evoluton and Averagng whch s done as follows.. Reconstructon step: Defne a pecewse polynomal ũx, t n ) from the current soluton u n. 2. Evoluton step: Use ũx, t n ) as an ntal condton for the gven dfferental equaton and compute ts soluton ũx, t n+ ) after one tme step t. 3. Averagng step: Determne the new cell average u n+ by computng the average of ũx, t n+ ) over the nterval C = x /2, x +/2 ), that s u n+ = x C ũx, t n+ )dx. The second step requres the soluton of a so-called Remann problem, whch s the hyperbolc equaton wth the dscontnuous ntal condton u l, for x x +/2, u x) = u r, for x > x +/2. The upwnd method for the advecton equaton s the result of ths method when usng the pecewse constant functon ũx, t n ) = u n, for all x C n the frst step so that u l = u n and u r = u n + n the Remann problem. Such an approach can obvously be mproved by usng a more precse reconstructon. Ths dea gves rse to the next secton.

22 22 Introducton to hyperbolc conservaton laws.3.3 The Lax-Wendroff method In [], Lax and Wendroff proposed the Lax-Wendroff method, whch s an mprovement of the upwnd method n terms of the order of consstency. To derve the method, begn wth the Taylor seres expanson of u to get ux, t n+ ) = ux, t n ) + tu t x, t n ) + 2 t2 u tt x, t n ) + O t 3 ). The tme dervatves can be replaced usng the advecton equaton.3) as u t + au x = u t = au x u tt = au x ) t = au t ) x = a 2 u xx. The O t) and O x) terms wll be omtted n the followng. Ths yelds ux, t n+ ) = ux, t n ) a tu x x, t n ) + 2 a2 t 2 u xx x, t n ). In a last step, approxmate the spatal dervatves wth central dfferences Together, ths results n u x x, t n ) = ux +, t n ) ux, t n ), 2 x u xx x, t n ) = u xx +/2, t n ) u x x /2, t n ) x = ux +,t n) ux,t n) x ux,t n) ux,t n) x x = ux +, t n ) 2ux, t n ) ux, t n ) x 2. ux, t n+ ) = ux, t n ) a t 2 x ux +, t n ) ux, t n )) + a2 t 2 2 x 2 ux +, t n ) 2ux, t n ) ux, t n )) so the Lax-Wendroff method for the advecton equaton has the form u n+ = u n a t ) u n 2 x + u n a t 2 ) + u n + 2u n + u n 2 x )..) Alternatvely, use a quadratc nterpolaton nstead of a lnear one as was done for the upwnd method. nterpolaton polynomal u n + + un + un x The data ponts x +, u n + ), x, u n ) and x, u n ) gve the x x + ) + un + 2un + un 2 x 2 x x + )x x ),

23 Introducton to hyperbolc conservaton laws 23 whch s the same as.) for x = x a t. The method can be wrtten n conservaton form snce u n+ = u n a t u n 2 x + u n a t ) + 2 x = u n t a u n x 2 + u n a ) 2 t 2 x wth = u n t x F n +/2 F n /2 ) ) 2 u n + 2u n + u n ) u n + 2u n + u n ) ) F /2 n = Fun, u n ) = a u n 2 + u n a ) 2 t u n 2 x u n ). Propertes of the Lax-Wendroff method and comparson wth the upwnd method. As mentoned before, the method s of second order. Analytcally t can be proven by the same approach as for the upwnd method. Begn wth Lx, t n ) = [ ux, t n ) ν t 2 ux +, t n ) ux, t n )) + ν2 2 ux +, t n ) 2ux, t n ) + ux, t n )) ux, t n+ ) Replace ux +, t n ), ux, t n ) and ux, t n+ ) agan by ther Taylor seres expanson ux ±, t n ) = ux, t n ) ± x u x x, t n ) + 2 x2 u xx x, t n ) ± 6 x3 u xxx x, t n ) + O x 4 ), ux, t n+ ) = ux, t n ) + t u t x, t n ) + 2 t2 u tt x, t n ) + 6 t3 u ttt x, t n ) + O t 4 ). The local truncaton error then becomes Lx, t n ) = [ ux, t n ) ν 2 x u x x, t n ) + ) t 2 3 x3 u xxx x, t n ) + ν2 2ux, t n ) + x 2 u xx x, t n ) + 2O x 4 ) 2ux, t n ) ) 2 ux, t n ) t u t x, t n ) 2 t2 u tt x, t n ) ] 6 t3 u ttt x, t n ) O t 4 ) = [ ν x u x x, t n ) ν6 t x3 u xxx x, t n ) + ν2 2 x2 u xx x, t n ) ν 2 O x 4 ) t u t x, t n ) 2 t2 u tt x, t n ) ] 6 t3 u ttt x, t n ) O t 4 ) = au x x, t n ) 6 a x2 u xxx x, t n ) + a2 2 tu xxx, t n ) a2 t x 2 O x4 ) u t x, t n ) 2 tu ttx, t n ) 6 t2 u ttt x, t n ) O t 3 ). ].

24 24 Introducton to hyperbolc conservaton laws In a last step, wrte the tme dervatves as space dervatves by means of the advecton equaton u t = au x u tt = a 2 u xx u ttt = a 3 u xxx, to get Lx, t n ) = 6 a x2 u xxx x, t n ) + 6 a3 t 2 u xxx x, t n ) + O x 3, t 3 ) = 6 aa2 t 2 x 2 )u xxx x, t n ) + O x 3, t 3 ). The method s n fact of second order, because the coeffcent of the u xxx term depends on t 2 and x 2. Fgure.8 proves ths as well, t shows the logarthmc error plot for the example that was used to llustrate the order of the upwnd method n the prevous secton. Lax Wendroff least squares approxmaton slope 2.6) error Fgure.8: The logarthmc error plot of the Lax-Wendroff method for the soluton of the advecton equaton wth an arc tangent as ntal condton. However, the Lax-Wendroff method does not necessarly gve much better results than the upwnd method. One dsadvantage s that the Lax-Wendroff method s not TVD as can be seen n fgure.9. There the upwnd method smoothes the dscontnutes and therewth compresses the functon whle the general form s conserved. Wth the Lax-Wendroff method ths s not the case and oscllatons are generated. Another dsadvantage s that the Lax-Wendroff method no longer has second order n such stuatons. The defnton and the proof of the order were based on the assumpton that the soluton s suffcently smooth. Ths s not gven for ths example and thus the theoretcal order of both the upwnd and the Lax-Wendroff method s no longer reached. Fgure. llustrates ths problem n

25 Introducton to hyperbolc conservaton laws 25 ntal condton Upwnd at tme t=.2 Lax Wendroff at tme t= Fgure.9: Soluton of the advecton equaton usng the upwnd and the Lax-Wendroff method. the form of the logarthmc error plots for the above example. Whle the order of the upwnd method s approxmately.6 nstead of, the order of the Lax-Wendroff method decreases by.3 to.7 and therefore s only margnally better than the one of the upwnd method..5 Upwnd.5 Lax Wendroff.5.5 least squares approxmaton slope.5954) error least squares approxmaton slope.78) error Fgure.: Logarthmc error plot for the upwnd and the Lax-Wendroff method.

26

27 2 The advecton equaton wth non constant velocty 2. Theoretcal analyss Consder now the advecton equaton as descrbed n secton.. wth the only dfference that the velocty a s no longer constant and equaton.3) takes the form ux, t) + ax) ux, t) =. 2.) t x Here ax) s restrcted to a pecewse constant functon of the form a, f x α, ax) = a 2, f x α wth < a a 2 and α an arbtrary value n space where the velocty changes. Ths generalzaton causes the characterstcs to no longer have the same slope as llustrated n fgure 2.. The slope s constant as long as the characterstcs do not cross the jump α where they knk. t α x Fgure 2.: Characterstcs for the advecton equaton wth pecewse constant velocty. Lke n the constant-coeffcent case, the soluton of the correspondng ntal value problem s constant along the characterstcs. For an arbtrary x α, they can be wrtten 27

28 28 2 The advecton equaton wth non constant velocty as Xt) = x + a 2 t, so that the problem reduces to the constant-coeffcent case. For x < α, the characterstcs can cross the poston of the jump, hence x + a t, f x + a t α t α x a Xt) = =: t, x + a t + a 2 t t ), else. The result of ths knk n some of the characterstcs and the dfferent slopes s that the ntal value u shfts to the rght as before but wth dfferent speed on each sde of the jump, so that the soluton dlates at the jump as shown n fgure 2.2. ntal condton, tme t= tme t= tme t= tme t= Fgure 2.2: Soluton of the advecton equaton wth pecewse constant velocty at dfferent tme steps. Here a =., a 2 = and the jump les at α =. These solutons can be computed by examnng the characterstcs at one pont. For the example n fgure 2.3 t holds that ux, t n ) = ux 2, t n+ ) and ux 5, t n ) = ux 6, t n+ ) for the solutons along the characterstcs that do not cross the jump and ux 3, t n ) = uα, t ) = ux 4, t n+ ) for the one that does.

29 2 The advecton equaton wth non constant velocty 29 t n+ x 2,t n+) x 4,t n+) x 6,t n+) t* t n x,t n ) x 3,t n ) x 5,t n ) Fgure 2.3: The soluton to the PDE s constant along each of the three possble shapes of the characterstcs. For the soluton ux, t) ths means that f x α, one smply has to follow the characterstc wth slope a untl t reaches tme t = where u s known through the ntal condton. The same holds for the characterstcs wth slope a 2 f x s suffcently bgger than α because f x α + a 2 t, the characterstc does not cross the jump. For values of x n between, one addtonally needs to determne the tme t where the characterstc crosses the jump, that s x 4 = α + a 2 t n+ t ) = x 3 + a t t n ) + a 2 t n+ t ) t = α + a 2t n+ x 4 a ) Overall, ths yelds the soluton as u x a t), f x α, ) ) ux, t) = u a a 2 α + a a 2 x a t, f α < x < α + a 2 t, u x a 2 t), f x α + a 2 t ) ) where u a a 2 α + a a 2 x a t = u α a t ) = u α a α+a2 t x a 2 )). It s easy to prove that ths s ndeed the soluton of the PDE by examnng the three cases. For x α, one obtans u t + a u x = u x a t)) t + a u x a t)) x = a u x a t) + a u x a t) =.

30 3 2 The advecton equaton wth non constant velocty The case x α + a 2 t s analogue wth a 2 nstead of a. For α < x < α + a 2 t, nsertng the soluton n the PDE results n u t + a 2 u x = u a ) α + a = a u =. a 2 a a 2 )) x a t a 2 t ) α + a ) x a t a 2 + a 2 u a a 2 a + a 2 u a a 2 a 2 ) α + a )) x a t a 2 ) ) α + a a 2 x a t x 2.2 Methods In ths secton, dfferent methods for solvng the advecton equaton wth pecewse constant velocty ax) = where a a 2 are ntroduced and analyzed. The a, f x α, a 2, f x α problem that arses when the velocty s no longer constant s that the CFL condton ν = ax) t x has stll to be fulflled. Snce the step sze t s generally constant, the spatal step sze x has to be chosen complyng the CFL condton for all values of ax). Alternatvely, the stencl of the method has to be adapted to ax). The followng sectons derve dfferent methods that are suted for the gven stuaton and analyze them theoretcally. At the end of the chapter n secton 2.3, they wll be compared wth each other by means of exemplary computatons The Upwnd method on a nonunform grd The most smple numercal method for solvng the advecton equaton s the upwnd method as ntroduced n secton.3.. Ths secton studes the ansatz of usng such a smple upwnd method wth the modfcaton of an adaptve spatal step sze to conform to the change n a. Consder a pecewse constant step sze x, f x α, x = x 2 = a 2 a x, f x α where x s the chosen value for the smaller step sze. Then t = ν x ax) = ν x a = ν x 2 a 2 wth ν, so that the CFL condton s satsfed.

31 2 The advecton equaton wth non constant velocty 3 Specal case To smplfy matters, assume for now that the spatal grd s constructed n such a way that the jump les on a grd pont, so that the characterstcs through the grd ponts do not cross the jump between tme steps. Ths results n the stuaton depcted n fgure 2.4. t n+ t n α x Fgure 2.4: The grd ponts and correspondng characterstcs on a nonunform grd wth courant number ν =. Snce the characterstcs do not knk durng a tme step and ν = a t x apply the upwnd method = a 2 t x 2, one can u n+ = u n νu n u n ) as n secton.3. on the nonunform grd wth the step szes x and x 2. Ths method was constructed so that t conserves the propertes of the upwnd method for the constant coeffcent advecton equaton. The proof that the method s of frst oder and TVD remans the same as n secton.3.. General case In a next step, consder an analogous method wthout the constrant that the jump les on a grd pont. There are two possble approaches to adjust the above method, whch are llustrated n fgure 2.5. Before, the jump separated the area wth small step sze from the one wth bgger step sze. Snce ths s no longer possble, the grd must be constructed n such a way that the jump les ether n an nterval of small step sze or n one of bgger step sze. As can be observed n the fgure, the jump n an nterval of smaller step sze x may result n the characterstc crossng a grd pont. Whether ths happens depends on the velocty a and the poston of the jump n relaton to the grd. The nearer the jump les

32 32 2 The advecton equaton wth non constant velocty t n+ t n+ t n α x t n α x Fgure 2.5: The grd ponts and correspondng characterstcs on a nonunform grd, where the jump s located n an nterval of sze x 2 left) or x rght). to the rght boundary of the nterval, the hgher s the possblty of the characterstc stayng n the nterval. Usng ths approach would therefore requre the method to make a case dstncton and then adjust the computatons accordngly. Ths can be avoded by usng the other approach. Creatng the grd so that the jump s n the bgger nterval of sze x 2 ensures that no characterstc through a pont x, t n+ ) crosses one spatal nterval n one tme step. Ths mples that the ordnary upwnd method can be used for all values except for u n+ where γ s the grd pont to the left of the jump wth γ = α x and α [x γ, x γ+ ). γ+ Dervaton of the numercal method The most smple method for ths case s to just keep usng the upwnd method and to gnore that the value u n+ γ+ s then computed wth the velocty a 2, whch s not correct because the velocty changes at the jump. In dong so, the propertes of the method reman as before. To mprove on ths, change the computaton of u n+ γ+. In secton.3. t was ponted out that the upwnd method can be nterpreted as a lnear nterpolaton. To compute a value u n+, nsert the pont x := x a t, where the characterstc through x, t n+ ) reaches t n, nto the nterpolaton polynomal to the data ponts x, t n ) and x, t n ). Ths can be done n ths stuaton as well. The thus derved method s then based on the dea of followng the characterstcs and s bascally no upwnd method anymore. At frst, compute the value x for ths stuaton. Equaton 2.2) stated that the characterstc through x γ+, t n+ ) passes the jump at tme t = a 2 α x γ+ + a 2 t n+ ).

33 2 The advecton equaton wth non constant velocty 33 Therefore, x = x γ+ a 2 t n+ t ) a t t n ) = α a t t n ) = a ) α + a x γ+ a t. 2.3) a 2 a 2 The value of the nterpolaton polynomal to x γ, t n ) and x γ+, t n ) at x s ) u n γ+ un γ+ un γ a + x x γ+ ) = u n γ+ a 2 α x γ+ ) a t + u n γ+ u n γ), x 2 x 2 whch s then the value of u n+ γ+. Thus, the method has the form u n+ = u n + a )α x a ) a t 2 x 2 u n un ), f = γ + for α [x γ, x γ+ ), u n νun un ), else. Note that f α + a 2 t x γ+, the characterstc through x γ+, t n+ ) does not reach the jump and therefore does not knk. In that case, t s not necessary to compute u n+ γ+ dfferently and all the propertes of the upwnd method are conserved as t was the case when the grd was constructed on the jump. Snce both approaches, the ordnary upwnd method and the modfed method, only dffer n the computaton of one value and have the same propertes, they yeld essentally the same results. The small dfference n u n+ γ+ has no notceable nfluence on the computaton of the values adjacent to t, so that ts part n the error s mnmal and, because of roundng n the computatons, not notceable The Upwnd method on a unform grd Consder now an upwnd method on a unform grd as an alternatve to the above examned upwnd method on a nonunform grd. The dea s to construct the grd unformly accordng to the smaller velocty a so that for gven x, t = ν x a wth ν. In ths case, the CFL condton may not be fullflled to the rght of the jump because a 2 t x = a 2 a ν and a 2 a for a a 2. Therefore, the method computes the soluton to the rght of the jump wth a bgger stencl. Under these condtons, the grd s constructed n such a way that to the left of the jump, the nterval between each of the ponts x and x at tme t n s only crossed by the characterstc that passes through x, t n+ ). To the rght of the jump, ths s stll the case n a way, but the characterstcs pass several spatal ntervals n one tme step snce

34 34 2 The advecton equaton wth non constant velocty the slope of the characterstcs to the rght of the jump s s := a 2 a tmes bgger than the one to the left. Assume here that s s an nteger, otherwse t has to be replaced by a2 a. Furthermore, some of the characterstcs now cross the jump durng a tme step, whch results n more than one characterstc passng the ntervals around the jump at tme t n. Specal case At frst, lmt the analyss to the case that the jump les on a grd pont lke n the prevous secton. Let β α be the smallest grd pont for whch the characterstc through β, t n+ ) does not reach the jump at tme t n and let β = α + s β. Denote further ᾱ, β and β the ndces of α, β and β, that s xᾱ = α, x β = β and x β = β where β = ᾱ + s and β = ᾱ + a 2 t x = ᾱ + sν. An example for ths stuaton s llustrated n fgure 2.6. Note that β = β f the Courant number s suffcently bg, so that sν = s. t n+ t n α β' β x Fgure 2.6: The grd ponts and correspondng characterstcs on a unform grd. Dervaton of the numercal method As for the nonunform grd, start wth usng the upwnd method and adjust the stencl accordngly. That means, compute u n+ = u n νu n u n j ) where the grd pont x j has yet to be determned. As the fgure suggests, there are three dfferent cases that have to be examned when computng the soluton at x, t n+ ). They are dstngushed from each other by the poston of x n relaton to α and β.

35 2 The advecton equaton wth non constant velocty 35 For x α, smply take x j = x as for the constant coeffcent advecton equaton. For the case x β, note agan that the slope of the correspondng characterstcs s s tmes bgger than for x α, so the stencl should be s tmes broader, whch mples x j = x s. Therefore, u n+ α depends on u n α and un+ β on u n α, so that u n+ should be computed wth u n α for α < x < β. Altogether, the upwnd method s u n+ = u νun un ), f ᾱ, u n νun un ᾱ ), u n νun un s ), f β. f ᾱ < < β, 2.4) As mentoned before, s should be replaced by a2 a f s s no nteger. Then, t could be bgger on the nterval wth broader stencl, so the coeffcent µ would change. Thus, fxng the coeffcent s a smplfcaton n ths case. A dstnctve feature of ths method les n the computatons for α < x < β snce they all use the value u n ᾱ. Ths results n a star-step lke pattern as llustrated n fgure 2.7. Fgure 2.7: Soluton usng the upwnd method. The rght fgure shows a magnfcaton of the dashed rectangle n the left fgure. Nevertheless, the method s convergent and the dscontnutes become smaller wth a

36 36 2 The advecton equaton wth non constant velocty smaller step sze. To confrm ths, examne the method for the TVD property. T V u n+ ) = = u n+ u n+ = = ᾱ u n+ u n+ + u n+ + u n+ β u n+ + β = β+ Compute each part separately, for the frst one ᾱ u n+ u n+ = = = ᾱ = ᾱ = ν) ᾱ+ un+ ᾱ + u n+ u n+. β =ᾱ+2 u n+ u n+ u n νu n u n ) u n + νu n u n 2) ν)u n u n ) + νu n u n 2) ᾱ = snce ν. For the second term, t holds u n+ In the thrd one, β =ᾱ+2 ᾱ+ un+ ᾱ u n+ u n+ u n u n + ν ᾱ = u n u n 2 = u nᾱ+ νu n ᾱ+ u n ᾱ ) u n ᾱ + νu n ᾱ u n ᾱ ) = ν) u n ᾱ+ u n ᾱ. = = β =ᾱ+2 β =ᾱ+2 ν) u n νu n u n ᾱ ) u n + νu n u n ᾱ ) u n u n νu n u n ) β =ᾱ+2 u n u n. The fourth part s gven as u n+ u β n+ = u β n β νu n β u n β s ) u n β + νu n β u n ᾱ ) ) = ν) u n β u n β + ν u n β s uᾱ ) n ν) u n β u n β u + ν n β s u n ᾱ

37 2 The advecton equaton wth non constant velocty 37 and the last as u n+ u n+ = = β+ Together, ths yelds T V u n+ ) ν) + ν ᾱ = = = β+ = β+ ν) = u n νu n u n s) u n + νu n u n s ) ν)u n u n ) + νu n s u n s ) = β+ u n u n u n u n + ν = β+ u n u n 2 + ν u n β s u n ᾱ + ν u n s u n s. = β+ so that the method s TVD f the last three summands add up to ν Snce β = ᾱ + s, ) := = = = ᾱ = ᾱ = ᾱ = = u n u n 2 + u n β s u n ᾱ + u n u n + u n ᾱ u n ᾱ + u n u n + u n ᾱ u n ᾱ + u n u n = β+ =ᾱ+s+ =ᾱ+ u n s u n s, = u n s u n s u n s u n s u n u n and t s guaranteed that the small dscontnutes do not become oscllatons. u n un. To be able to compute a soluton wthout ths feature, try mprovng the method by usng the same ansatz as was done for the nonunform grd. Case. For the soluton at x α, smply take the upwnd method as ntroduced n secton.3., that s u n+ = u n a t x un u n ), f ᾱ. 2.5)

38 38 2 The advecton equaton wth non constant velocty Use now a lnear nterpolaton to derve the method for x > α as well. For the nterpolaton error to be mnmal, the data ponts have to be chosen as close as possble to the pont where the polynomal s evaluated, that means the grd ponts drectly to the left and rght. For x α, ths was gven by takng x, t n ) and x, t n ), whch s no longer the case for x > α. Case 2. The characterstcs through all the ponts x, t n+ ) wth α < x < β end between xᾱ and xᾱ at tme t n. Therefore, the method should compute the soluton at x, t n+ ) by means of xᾱ, t n ) and xᾱ, t n ). The nterpolaton polynomal to ths two data ponts s u n ᾱ + un ᾱ u n ᾱ x xᾱ) = u n ᾱ + x α xᾱ xᾱ x un ᾱ u n ᾱ ). To determne x, observe that the characterstcs through xᾱ+j, t n+ ) wth j =,..., β ᾱ pass the jump at tme t n+ j t wth a constant tme nterval t. It holds that α + a 2 j t = xᾱ+j j =,..., β ᾱ t = x a 2, so that for the characterstc through x, t n+ ) wth j = ᾱ x = α a t j t) = α a t j x ) = α a t + ᾱ) x a 2 s. In the nterpolaton polynomal, ths gves so that u n ᾱ + a t + ᾱ) x s u n ᾱ u n x ᾱ ) = u n ᾱ + ᾱ s a ) t u n ᾱ u n x ᾱ ) ) ᾱ u n+ = u n ᾱ + ν u n ᾱ u n s ᾱ ), f α < x < β. 2.6) Case 3. To the rght of the jump, the exact soluton at x, t n+ ) wth x β depends on x a 2 t, t n ). The pont x a 2 t = x sν x les between the grd ponts x sν x = x sν and x sν +, so a numercal method should compute the soluton usng these two ponts. Usng a lnear nterpolaton through x sν and x sν + yelds the nterpolaton polynomal u n sν + un sν + un sν x x x sν + x sν ) = u n sν + un sν + un sν x x sν x sν ). Insertng x = x sν x results n u n sν + x sν x x sν u n sν + x un sν ),

39 2 The advecton equaton wth non constant velocty 39 so the soluton s u n+ = u n sν + sν sν)un sν + un sν ), f β. 2.7) Combnng the solutons 2.5), 2.6) and 2.7) of the three cases, the method can then be wrtten as u n+ = u n νun un ), f ᾱ, u n ᾱ + ᾱ s ν ) u n ᾱ u n ᾱ ), f ᾱ < < β, u n sν + sν sν)un sν + un sν ), f β. 2.8) Whle the approach of followng the characterstcs to modfy the upwnd method changed the method for the nonunform grd only at one pont, the method for the unform grd s altered much more. Ths has the advantage of yeldng a more accurate soluton wthout the addtonal dscontnutes that where generated n the example of fgure 2.7. The proof that ths method s TVD can be found n the appendx. General case Relax now the restrcton that the jump les on a grd pont and modfy the above methods accordngly. Dependng on the poston of the jump n relaton to the grd ponts, the characterstcs that cross the jump n a tme step end ether n the nterval of the jump or the one to the left. Fgure 2.8 shows both possbltes and t can be observed that the farther to the rght of the nterval the jump les, the more characterstcs stay n the nterval of the jump. t n+ t n+ t n α β' β x t n α β' β x Fgure 2.8: The grd ponts and correspondng characterstcs on a unform grd. Keep the notaton of before, that s s = a 2 a. Snce α s now no grd pont, let γ agan be the grd pont to the left of the jump, so that γ = α x and α [x γ, x γ+ ). Furthermore, adjust the values of β and β as β = γ + s + and β = α+a 2 t x = α x + sν.

40 4 2 The advecton equaton wth non constant velocty Dervaton of the numercal method The smple upwnd method s the same as 2.4) wth ᾱ substtuted by γ, that s u n νun un ), u n+ = u n νun un γ ), u n νun un s ), f γ, f γ < < β, f β. Snce ths method s essentally the same as method 2.4), t has the same propertes and creates the same star-shaped dscontnutes as depcted n fgure 2.7. Prove agan that these small dscontnutes do not oscllate for smaller Courant number by verfyng the TVD property. The frst part of the proof s the same as before f ᾱ s substtuted by γ. Dfferences occur when provng that ) := γ = u n u n 2 + u n β s u n γ + = β+ because now γ = α x and β = γ + s +, so that ) = = γ = γ = = u n u n + u n γ+ u n γ + u n s u n s = = γ+s+2 u n u n + u n γ+ u n γ + u n γ u n γ + u n u n. = u n s u n s = γ+2 u n u n u n u n, Consder now agan the modfed upwnd method. For x γ and x β, the value u n+ can be computed n the same way as derved n case and 3 before, only the case γ < x < β has to be treated dfferently. The easest but therefore less accurate ansatz s to make a lnear nterpolaton as before and use the data ponts x γ, u n γ ) and x γ+, un γ+ ) to cover both ntervals [x γ, x γ ) and [x γ, x γ+ ) where the characterstc through x, t n+ ) mght reach the pont x, t n ). In secton 2.2., x was already determned as x = a ) α + a x a t. a 2 a 2 The nterpolaton polynomal s u n γ+ + un γ+ un γ x x γ+ ) = u n γ+ + x x γ+ u n γ+ u n γ ), x γ+ x γ 2 x

41 2 The advecton equaton wth non constant velocty 4 so that u n+ = u n γ+ + x x γ+ u n γ+ u n γ ) 2 x ) a = u n γ+ a 2 α + a a 2 x a t x γ+ + u n γ+ u n γ ) 2 x = u n γ+ + ) α 2 s x + ) s ν γ u n γ+ u n γ ), whch overall yelds u n νun un ), f γ, u n+ = u n γ+ + 2 α s) x + s ν γ ) u n γ+ un γ ), f γ < < β, u n sν + sν sν)un sν + un sν ), f β. As a somewhat more effcent approach, one could as well determne beforehand, for whch γ, β ) the value u n+ should be computed va x γ, u n γ ) and x γ, u n γ), or va x γ, u n γ) and x γ+, u n γ+ ). Ths depends on x, f x < γ, use the frst par and f x > γ, the second. For x = γ, both yelds the same result. These nequaltes can be solved to get x = a ) α + a x a t < γ a 2 a 2 x < sγ + sa t + s)α < s γ + sν + s) α x. The same computatons as before then gve u n νun un ), f γ, u n+ u n γ + α = s) x + s ν γ) u n γ u n γ ), f γ < s γ + sν + s) α x, u n γ + α s) x + s ν γ) u n γ+ un γ), f s γ + sν + s) α x < < β, u n sν + sν sν)un sν + un sν ), f β. 2.9) As for the specal case above, the proof of the TVD property for ths method s n the appendx The Lax-Wendroff method The last two sectons provded dfferent, smple methods for the soluton of the advecton equaton wth non constant velocty and therefore ensure that a numercal soluton can

42 42 2 The advecton equaton wth non constant velocty be computed. In the followng, these results shall be mproved n the sense of ganng a method of hgher order. As before, derve the Lax-Wendroff method and adjust t to the two cases of a nonunform and unform grd by followng the characterstcs. The approach of usng the ordnary Lax-Wendroff method wll not be examned further snce t can be derved smlarly to the Upwnd method. Furthermore, the am s to gan a method that s as accurate as possble and even for the Upwnd method, the modfed method yelded somewhat better results. Nonunform grd Consder the stuaton of secton 2.2. now wthout the assumpton that the jump les on a grd pont, that s α [x γ, x γ+ ) wth γ = α x. For all / { γ, γ + }, the stencl of the Lax-Wendroff method les n an nterval of constant step sze, so that the Lax-Wendroff method of secton.3.3 can be used. When computng the value u n+ γ, only the dfferent step szes have to be consdered. Interpolate usng the data ponts x γ+, t n ), x γ, t n ) and x γ, t n ) to get the polynomal u n γ+ + u n γ+ u n γ) x x γ+ u n γ+ u n γ + x 2 x 2 where x = γ a t = γ ν x, so that u n+ γ = u n γ+ u n γ+ u n γ) ν x + x 2 x 2 u n γ+ u n γ + un γ u n γ x 2 x = u n γ u n γ+ u n γ) ν x x 2 + u n γ+ u n γ x 2 un γ u n γ ) x x γ+ )x x γ ) x x + x 2 ) ν x + x 2 )ν x x + x 2 un γ u n γ x ) ν x + x 2 )ν x x + x 2. For u n+ γ+, the Lax-Wendroff method uses the data ponts x γ+2, t n ), x γ+, t n ) and x γ, t n ), so that the step sze between these ponts s x 2, but now the characterstc crosses the jump. The nterpolaton polynomal s then the same as n secton.3.3, namely and x = u n + + un + un x 2 a a 2 ) x x + ) + un + 2un + un 2 x 2 x x + )x x ), 2 α + a a 2 x a t wth = γ + as n the other stuatons before where the characterstc crossed the jump. Together u n+ γ+ = un γ+2 + x x γ+2 u n γ+2 u n γ+ ) x x γ+2 )x x γ+ ) + u n γ+2 x 2 2 x 2 2u n γ+ + u n γ) 2

43 2 The advecton equaton wth non constant velocty 43 where x x γ+2 = x x γ+ x 2 and x x γ+2 )x x γ+ ) = x x γ+ ) 2 x 2 x x γ+ ). Insertng ths yelds u n+ γ+ = un γ+2 u n γ+2 u n γ+ ) x x γ+ + u n γ+2 u n γ+ u n γ+2 2u n γ+ + u n γ ) ) x x x γ+ ) 2 u n γ+2 2u n γ+ + u n γ) 2 x 2 2 = u n γ+ + x x γ+ 2 x 2 = u n γ+ + + u n γ+2 u n γ ) x x γ+ ) 2 + u n γ+2 2 x 2 2u n γ+ + u n γ) 2 a a 2 ) α x γ+ ) ν x 2 x 2 ) ) 2 a a 2 α x γ+ ) ν x 2 x 2 2 u n γ+2 u n γ ) u n γ+2 2u n γ+ + u n γ ). Overall, the Lax-Wendroff method for a nonunform grd s then defned as u n+ = u n un + un ) ν x x 2 u n + + u n + u n x 2 + a un un x ) ν x + x 2 )ν x x + x 2, f = γ, a 2 )α x ) ν x 2 x 2 ) ) a 2 α x a ) ν x 2 2 x 2 2 u n ν 2 u n + u n ) + ν2 2 u n + u n ) u n + 2u n + ) un, f = γ +, u n + 2u n + ) un, else. Unform grd The dfference between the upwnd method and the Lax-Wendroff method on a unform grd s the hgher order nterpolaton. Snce the step sze x s now constant, the nterpolaton polynomal for three arbtrary data ponts x, t n ), x, t n ) and x +, t n )

44 44 2 The advecton equaton wth non constant velocty has the form u n + + u n + u n ) x x + x = u n + + u n + u n ) x x x = u n + x u n + u n ) 2 un + 2u n + u n ) + u n + 2u n + u n ) x x + )x x ) 2 x 2 + u n + 2u n + u n ) x x ) 2 xx x ) ) 2 x 2 x x + u n + 2u n + u n x ) x x ) 2 2 x 2 = u n + u n + u n ) x x 2 x + un + 2u n + u n ) x x ) 2 2 x 2. Accordng to the cases examned n secton 2.2.2, the correspondng data ponts and x α x have to be nserted. Note agan that α [x γ, x γ+ ), γ = and β = α+a 2 t x = α x + sν wth s = a 2 a. The frst case, γ, s not affected by the jump, so that the ordnary Lax-Wendroff method wth x, t n ), x, t n ), x +, t n ) and x = x a t can be used wth u n+ = u n ν u n 2 + u n ν ) 2 + u n 2 + 2u n + u n ). The second case, γ < < β, was subdvded nto the two cases where the ntervals [x γ, x γ ] and [x γ, x γ+ ] were dfferentated. Ths s no longer necessary when usng three data ponts for the nterpolaton as both ntervals are covered by one nterpolaton polynomal. Thus, ) choose x γ, t n ), x γ, t n ) and x γ+, t n ), whle the pont x remans x = a a 2 α + a a 2 x a t as before. The result s then u n+ = u n γ + u n γ+ u n γ ) x x γ 2 x + un γ+ 2u n γ + u n γ ) x x γ ) 2 2 x 2 = u n γ + a ) α 2 a 2 x + ) s ν γ u n γ+ u n γ ) + a ) α 2 a 2 x + ) 2 s ν γ u n γ+ 2u n γ + u n γ ). In the thrd case, β, the hgher velocty had to be consdered, so the data ponts x sν, t n ) and x sν +, t n ) and x = x a 2 t = x sν x were used. Now addng x sν, t n ) yelds u n+ = u n sν + un sν + un sν )x x sν 2 x + u n sν + 2un sν + un sν )x x sν ) 2 2 x 2 = u n sν + 2 sν sν)un sν + un sν ) + 2 sν sν)2 u n sν + 2un sν + un sν ).