c 2001 Society for Industrial and Applied Mathematics

SIAM J. SCI. COMPUT. Vol., No. 6, pp. 56 81 c 001 Society for Idustrial ad Applied Mathematics HIGH RESOLUTION SCHEMES FOR CONSERVATION LAWS WITH LOCALLY VARYING TIME STEPS CLINT DAWSON AND ROBERT KIRBY Abstract. We develop upwid methods which use limited high resolutio correctios i the spatial discretizatio ad local time steppig for forward Euler ad secod order time discretizatios. L stability is prove forboth time steppig schemes forproblems i oe space dimesio. These methods are restricted by a local CFL coditio rather tha the traditioal global CFL coditio, allowig local time refiemet to be coupled with local spatial refiemet. Numerical evidece demostrates the stability ad accuracy of the methods for problems i both oe ad two space dimesios. Key words. spatially varyig time steps, upwidig, coservatio laws AMS subect classificatios. 35L65, 65M1, 65M30 PII. S106487500367737 1. Itroductio. Hyperbolic coservatio laws model a wide rage of physically iterestig pheomea such as gas dyamics, shallow water flow, ad advectio of cotamiats. Coservative high resolutio methods with explicit time discretizatio have prove effective i capturig the sharp, movig frots commo i these applicatios. It is well kow that such methods require the time step to satisfy a CFL coditio i order to guaratee stability. Local spatial refiemet is ofte itroduced i order to resolve these frots more efficietly. However, this local refiemet reduces the allowable time step for the explicit time discretizatios typically employed. Rather tha cosiderig a fully implicit approach, i which the step size is ofte costraied by the oliear covergece ayway, we cosider a method which allows the time step to vary spatially ad satisfy a local CFL coditio. I this way, we ca icrease the efficiecy of the time steppig sigificatly i certai situatios. Aother situatio where local time steppig is useful is whe modelig trasport of some quatity by a highly varyig velocity field. The advectio of a species is described by a equatio of the form 1) c t + uc) =q, where c is the cocetratio of the species, u is the velocity of the fluid trasportig the species, ad q represets source/sikterms. High resolutio schemes have become popular for these problems, primarily because they are coservative ad satisfy a maximum priciple. The CFL costrait o the time step is determied by the ratio of the mesh spacig ad the magitude of the velocity. I certai cases, specifically i the presece of iectio or productio wells, the magitude of u ca vary substatially throughout the domai. I these situatios, eve if the mesh spacig is uiform, usig a global CFL time step ca be very restrictive, ad local time steppig ca result i substatial savigs i computatio time. It is importat that the Received by the editors February 14, 000; accepted for publicatio i revised form) October 30, 000; published electroically April 6, 001. This work was supported by NSF grat DMS-9805491. http://www.siam.org/ourals/sisc/-6/36773.html CeterforSubsurface Modelig C000, Texas Istitute forcomputatioal ad Applied Mathematics, Uiversity of Texas at Austi, Austi, TX 7871 clit@ticam.utexas.edu, rob@ticam. utexas.edu). 56

HIGHER ORDER LOCAL TIME STEPPING 57 local time steppig preserve the coservatio ad maximum priciple properties of the uderlyig method. This situatio was explored i some detail from a practical viewpoit i [3], ad we will explore it from a more theoretical viewpoit here. Local time steppig for oe-dimesioal scalar coservatio laws was first proposed i Osher ad Saders [11]. They gave a thorough aalysis of a first order spatial discretizatio with a local forward Euler time steppig scheme. This scheme allows each elemet to take either a etire time step or some fixed M smaller steps. I [3], the first author examied local time steppig for advectio equatios of the form 1), usig high resolutio schemes with slope limiters. Numerical results i two space dimesios were preseted. Aother approach to local time steppig ivolves automatically takig smaller time steps where the mesh is refied. Berger ad Oliger developed such a approach i [1], i which refied grids are laid over regios of the coarse mesh. These fie grids ca have differet orietatio tha the coarse oe, ad eed ot be ested. Whe the problem is itegrated i time, small time steps are take o the refied mesh ad large time steps o the coarse mesh. Iformatio is the passed betwee the grids by meas of iectio ad iterpolatio. This approach allows higher order time itegratio, as the computatio for each time step is doe idepedetly oce the iformatio is passed at the begiig of the time step. I other work, Kallideris ad Baro [8] coupled the time refiemet to the spatial refiemet but used a sigle ouiform mesh rather tha multiple meshes. This work preseted several methods all first order) of hadlig the iterface betwee regios with differet time steps. Flaherty et al. [5] have developed a parallel, adaptive discotiuous Galerki method with a local forward Euler scheme which relies o iterpolatig values i time at iterfaces betwee time steps of differet sizes. This scheme, however, does ot appear to coserve flux alog these iterfaces. Also, oly first order i time methods are discussed. I this paper, we seekto exted the worki [11] ad [3] i two ways. First, we will show a maximum priciple for a local forward Euler method whe limited slopes are icluded. Secod, we will show that the mai ideas of local forward Euler discretizatios may be exteded to secod order i time by way of the TVD Ruge Kutta methods of Gottlieb ad Shu [6] ad Shu [13]. A stability result for a costat coefficiet case will be derived for this higher order method. We will also idicate how to exted this result to oliear problems. This aalysis shows that we oly eed to satisfy a local CFL coditio o each elemet. The paper is outlied as follows. First, i sectio, we formulate a high resolutio local forward Euler time discretizatio that employs some limited correctio to the piecewise costat solutio. I sectio.1, a maximum priciple is derived for this approach. The, a local time steppig procedure based o a secod order time discretizatio is described i sectio 3, ad a maximum priciple for a simple case is give i sectio 3.1. Some extesios ad implemetatio details are also discussed i this sectio. Fially, umerical results validatig the theory are preseted i sectio 4.. A high resolutio method. We first cosider the scalar coservatio law ) c t + f c) x =0, together with the iitial coditio 3) cx, 0) = c 0 x).

58 CLINT DAWSON AND ROBERT KIRBY We will partitio the real lie ito itervals I = {x : x 1 x<x + 1 }. We use the differece operator + to deote a forward differece. That is, + c c +1 c for some quatity c. A similar defiitio holds for with c c c 1. By itegratig ) over each I ad usig some cosistet, Lipschitz umerical flux h at the cell edges, we obtai the semidiscrete scheme dc 4) dt = 1 + hc 1,c ), x where c is a approximatio to the itegral average of c over I. Examples of h iclude the Goduov flux ad the Lax Friedrichs flux. It is importat that h is odecreasig i the first variable ad oicreasig i the secod variable. The iitial coditio is defied by simple proectio. A stadard forward Euler discretizatio ca also be obtaied i the obvious way. I order to improve the accuracy of the above approach, we icorporate higher order correctios ito the method. There are several ways of costructig these correctios. These iclude the piecewise liear recostructio used i the mootoe upwid scheme for coservatio laws MUSCL) of va Leer [14], the ENO recostructio schemes of Harte ad Osher [7], ad the Ruge Kutta discotiuous Galerki methods of Cockbur ad Shu []. We shall ot deal exclusively with ay specific method but require certai properties of whichever is used. Sice we are iterested i secod order accuracy, all of our examples assume liear approximatios i space i each iterval I. After costructig the correctios ad limitig them, we add them to the meas to costruct more accurate left ad right states for the Riema solutios. The correctios are deoted by tildes. Left states at a iterface x +1/ shall be deoted as c + c c + 1,L ad right states as c +1 c +1 c + 1,R. I this way, the umerical flux at the edge is give by hc + 1,L,c + 1,R ). Vital to the aalysis is the assumptio that the correctios itroduce o ew extrema ito the solutio. This allows correctios derived from the mimod slope limiter described below) but rules out those derived from the modified mimod limiter of Shu [1] ad the ENO schemes as itroduced by Harte ad Osher [7]. We ca quatify this restrictio as 5) ad θ + c + c 1 θ + c 6) 1, + c where θ 0. The umber θ is a chose parameter which appears i the CFL coditio. For example, see []. Geometrically, θ is related to the largest allowable ratio of differeces betwee left ad right states at cosecutive edges relative to the meas. This ratio ca be allowed to be larger tha 1, ad we will see that such a situatio will ecessitate a restrictio of the time step i order to guaratee a maximum priciple. Cosider the cells ad + 1 i Figure 1. The value c + 1,L c 1,L c +1 c ; however, the differeces have the same sig. Thus 7) 0 c + 1,L c 1,L =1+ + c, c +1 c + c

HIGHER ORDER LOCAL TIME STEPPING 59 c -1/,L c +1/,L -1 +1 Fig. 1. Slopes icrease ratio of left states to meas. ad if θ satisfies 6), the 0 c + 1,L c 1 8),L 1+θ. c +1 c Specifically, suppose that we compute a slope, δc, usig the mimod limiter. Thus, 9) { +c mi δc = +x 0 otherwise., c x ) sg + c ) if sg + c ) = sg c ), Set c u = x δc. The set 10) c = { c u if c u ±c, 0 otherwise. This last limitig step is ecessary oly i the case of ouiform mesh. c = c. The correctios the have the property that ) c 11) sg 0. ± c We ca thus make the boud Fially, 1) We also have 13) 1+ + c + c 1+ c +1 + c. 1+ + c 1 c 0 + c + c

60 CLINT DAWSON AND ROBERT KIRBY ad 14) 1 + c + c 1 c +1 + c 0. Hece, 5) ad 6) hold automatically with the choice θ =1. More geerally, suppose correctios c u ad c u are computed by some meas, ad θ is chose such that 0 θ 1. Defie { c u c = if c u θ ±c, 15) 0 otherwise, with a aalogous defiitio for c. The 16) ad 1+ + c + c 1 c +1 + c c + c 1 θ 0 1+ + c 17) 1+θ. + c We ow preset the first order local time discretizatio. It is a straightforward extesio of the method i [11]. The oly differece is that the flux terms are computed usig the corrected quatities c +1/,L, c +1/,R rather tha piecewise costats. We allow each elemet to take either a whole time step or some fixed M substeps per mai time step. We begi by otig that the itervals I form a partitio of the real lie. I additio to the spatial partitio, we also itroduce the temporal partitio of [0,T) ito time itervals [t,t +1 ), =0,...,N 1, with t 0 = 0 ad t N = T. We deote t t +1 t ad defie λ t x. I order to describe the local time steppig scheme, we will eed to further partitio the time steps o certai elemets. We deote by C the set of all idices such that a sigle time step is take from t to t +1 o I. O the rest of the elemets, we partitio the time step [t,t +1 )itothe uio of substeps [t +η l,t +η l+1 ), l =0,...,M 1. Further, {σ k } M k=1 is a sequece of positive umbers summig to uity. The umbers η l are give as the cumulative sum of the σ k, that is, η l = l k=1 σ k, ad η 0 = 0. Correspodigly, the substeps i the time iterval are give by t +η l+1 = t +η l + σ l+1 t. Notice that the elemets o which the local steps are take may chage over time. With this otatio established, we ca modify the predictor-corrector scheme of Osher ad Saders to a so-called high resolutio scheme i space. Followig [11], for each k =1,...,M 1, the predictor is defied by 18) c +η k = { c, C, c k 1 λ l=0 σ l+1 + hc +η l 1/,L,c+η l 1/,R ), / C, ad the corrector is 19) = c λ M 1 l=0 σ l+1 + hc +η l 1,L,c+η l 1,R). Notice that if 1,,+1 C, the the method o I reduces to forward Euler with a step size of t.

HIGHER ORDER LOCAL TIME STEPPING 61.1. A maximum priciple. I this sectio, we preset a maximum priciple for the method 18) 19). The maximum priciple will verify that L stability is attaiable with oly local assumptios regardig the mesh, time step, velocity, ad correctios. It is vital to this aalysis that we use a strict limiter i.e., oe that does ot itroduce ew extrema). We itroduce some otatio for the scheme as follows. We defie terms ivolvig the differeces of the correctios relative to the meas: 0) κ,1 1 c c ad 1) κ, 1+ c c, where we may have a superscript of, + η l, etc. Note that 5) ad 6) give us the bouds 0 κ,1,κ, 1+θ. We also defie the coefficiets γ,1 ad γ, as the local Lipschitz coefficiets of h. That is, ) γ +η l,1 Λ +η l ) h c +η l l h + 1,L,c+η + 1,R c +η l + 1,R c+η l 1,R c +η l + 1,L,c+η l 1,R ) ad 3) γ +η l, Λ +η l ) h c +η l l h c +η l l + 1,L,c+η 1,R 1,L,c+η c +η l + 1,L c+η l 1,L 1,R ), where 4) { Λ +η l λ = if 1, or +1 C, σ l+1 λ otherwise. If the flux f is smooth eough, the we ca, as i [11], reduce the Lipschitz costats to a local supremum of f c). Further, we defie the terms 5) α,1 γ,1 κ,1 ad 6) α, γ, κ,. By our assumptios 5) ad 6) ad the assumed mootoicity properties of h h odecreasig i its first argumet, oicreasig i its secod), we have α,1 0, α, 0, ad thus 7) α,1 α, 0. For each at each time t +η l, we require the local CFL coditio 8) 1 α +η l,1 + α +η l, 0.

6 CLINT DAWSON AND ROBERT KIRBY Here, we pause to discuss the relatio betwee the terms 5) 6) ad the CFL coditio 8). We require that 9) α +η l,1 α +η l, 1, or 30) γ +η l,1 κ +η l,1 γ +η l, κ +η l, 1. If the correctios are all zero, the each κ term is simply 1. However, as i Figure 1, our κ terms are oly bouded by 1+θ. Cosequetly, i order to satisfy 8), we must possibly take a smaller time step tha we would if we were ot usig the correctios. Equivaletly, we could write our CFL coditio as do Cockbur ad Shu []: 31) γ +η l,1 γ +η l, 1 1+θ. The L stability result of [11] depeds upo showig that the operator which advaces the solutio forward is mootoe. However, with the additio of the correctio terms, the operator ceases to be mootoe, ad we must take a differet approach. We begi by examiig the basic global time steppig method or C ). The method is 3) = c λ [ ) )] h c + 1,L,c+ 1,R h c 1,L,c 1,R. We ca obtai L stability as follows. Add ad subtract hc + 1,L,c 1,R) i 3) ad multiply ad divide by the proper differeces, ad we have 33) = c + α,1 c +1 c ) + α, c c 1), or 34) = 1 α,1 + α,) c + α,1c +1 α,c 1. The, by 7) ad 8), we see that 35) 1 α,1 + α,) c + α,1 c +1 α, c 1 sup c. By makig similar argumets i the cotext of the local time steppig method, we have the followig stability result. Propositio.1. Assume 5), 6), ad 8) hold, ad the umerical flux hu, v) is odecreasig i u ad oicreasig i v; the for >0, sup c sup c 0, where c is defied by 18) 19). Proof. The proof follows from lookig separately at the cases i which C ad / C.

HIGHER ORDER LOCAL TIME STEPPING 63 We begi with the case of / C. I this case, we may use the same techique as i the global time steppig case recursively to boud the solutio at time +1i terms of the solutio at time. 36) c +η l = = c +η l 1 c +η l 1 + α +η l 1, = c +η l 1 σ l+1 λ + h + α +η l 1,1 c +η l 1 ) c +η l 1 1,L,c+η l 1 1,R ) c +η l 1 +1 c +η l 1 ) c +η l 1 1 1 α +η l 1,1 + α +η l 1, + α +η l 1,1 c +η l 1 +1 α +η l 1, c +η l 1 1 max 1 k +1 c+η l 1 k. The α terms satisfy the same sig coditios as their global time steppig couterparts, so ) 37) c +η l max 1 k +1 c+η l 1 k. If [ 1, + 1] is the elemet o which the maximum occurs, ad if C, the we are doe for this elemet, as we have the solutio bouded i terms of values at time. If / C, the we apply this argumet recursively. Now, suppose that C. Usig 18) i 19), 38) = c λ = = = M 1 l=0 M 1 l=0 M 1 l=0 M 1 l=0 ) σ l+1 + h c +η l l 1,L,c+η 1,R σ l+1 c +η l 1 λ + h )) c +η l l 1,L,c+η 1,R [ ) σ l+1 c +η l 1 + α +η l 1,1 c +η l 1 +1 c +η l 1 + α +η l 1, c +η l 1 )] c +η l 1 1 [ ) σ l+1 c +η l 1 1 α +η l 1,1 + α +η l 1, ] + α +η l 1,1 c +η l 1 +1 α +η l 1, c +η l 1 1. Agai, usig the mootoicity of h ad 8), we have the sig coditios for the α terms, ad 39) M 1 [ ) σ l+1 c +η l 1 1 α +η l 1,1 + α +η l 1, l=0 ] + α +η l 1,1 c +η l 1 +1 α +η l 1, c +η l 1 1 M 1 l=0 σ l+1 max 1 k +1 c+η l 1 k.

64 CLINT DAWSON AND ROBERT KIRBY Now, each term c +η l 1 k may be bouded by prior iformatio. If k C, the c +η l 1 k = c k ad the boud is obvious. Otherwise, the above boud for / C applies, ad we are doe. Note that while c remais costat i time o I, we might have to modify the correctio terms o I at itermediate times if C ad 1 / C or +1 / C to avoid addig extrema to the solutio. Note also that for geeral problems, the CFL coditio 8) for substeps after the first whe / C caot be verified a priori without adustig the time step at each substep. However, examiig 5) ad 6), we have uiform bouds o the κ terms by doig appropriate limitig. Moreover, with some kowledge of f, we ca boud the γ terms locally depedig o f c) ad the local mesh spacig. Thus we ca estimate a upper boud o the α terms over regios of the domai, which ca be used to set the local time steps over these regios. We have used this approach successfully i practice. 3. A secod order time steppigscheme. We ow tur to developig a local time steppig procedure based o a formally secod order method. We are agai iterested i itegratig the semidiscrete scheme 40) dc dt = 1 x + hc 1/,L,c 1/,R ). We first describe the basic method we are usig, which is a secod order Ruge Kutta scheme i particular, Heu s method), show to be TVD i [13, 6]. The, we formulate the scheme for hadlig the iterface betwee two regios with differet time steps. We will assume the basic method holds for some distace away from the iterfaces. Heu s method for itegratig 40) is give by 41) = c λ + hc 1/,L,c 1/,R ), 4) w +1 = λ + h 1/,L, c+1 1/,R ), 43) = 1 c + w +1 ). Here +1/,L = c+1 +, with a aalogous defiitio for +1/,R, where the correctios are computed as i 5) 6). This method is othig more tha a covex combiatio of forward Euler steps ad the iitial value. Correspodigly, usig the properties of the forward Euler method aalyzed i the previous sectio, oe easily obtais stability for this scheme. Propositio 3.1. For the scheme 41) 43), with correctios limited as i 5) 6) ad a CFL time-stepcostrait as i 8), we have sup c sup c 0. For ow, we are iterested i computig at the iterface o the space-time mesh show i Figure. Thus, the time step i iterval I +1 is t ad i I, t/. For

HIGHER ORDER LOCAL TIME STEPPING 65 f +1-1/ f +1 f +1 +1/ +1 +3/ +1 c +1 f +1/ -1/ f +1/ -1/ / / f +1/ +1/ f +1/ +1/ f -1/ c f +1/ c +1 f +3/ Fig.. Local time steppig mesh. simplicity, we assume the time step is t for itervals to the right of x +1/ ad t/ for itervals to the left of x +1/. We will also assume for simplicity that fc) =uc, where u>0. We will describe the method ad state coditios which the correctios c should satisfy to esure a maximum priciple. We will the describe i more detail how the correctios may be costructed so as to satisfy these coditios. Fially, we will describe the method whe the situatio i Figure holds ad u<0, ad the discuss geeralizatios to a oliear flux fuctio ad to the case with M steps. Thus, assumig fc) =uc, where u>0, to compute the solutio i elemets I ad I +1, we use the followig procedure. Scheme I u >0). i) Compute correctios c 1, c, ad c,1 +1 by some meas ad limit them so that 44) is satisfied. ii) c + 1 = c uλ [ c + c )] c 1 + c 1. iii) Compute / 1, /, ad c, +1 so that 45) is satisfied. iv) w + 1 = c + 1 uλ 1 [c+ + c + 1 c + 1 1 + c + 1 1 )]. v) c + 1 = 1 c + 1 w+ ). vi) Defie c +1 1 c,1 +1 + c, +1 ). vii) +1 = c+1/ +1 uλ +1 [c +1 + c +1 1 c + c + 1 c+ + c + 1 )]. viii) Compute c + 1 1 1, c+, ad,1 +1 so that 46) is satisfied. iv) = c uλ 1 [c+ + c + 1 c + 1 1 1 + c+ 1 )]. x) Compute 1,, ad, +1 so that 47) is satisfied. xi) w +1 = uλ [c+1 + 1 + 1 )]. xii) = 1 1 c+ + w +1 ). xiii) Defie +1 1,1 c +1 + c, +1). xiv) w +1 +1 = c +1 uλ +1 [c +1 + c +1 1 1 c+ + c + 1 + + )]. xv) +1 = 1 c +1 + w +1 ) +1.

66 CLINT DAWSON AND ROBERT KIRBY The correctios should satisfy the followig coditios to guaratee a maximum priciple: 44) θ c c 1 c, c,1 +1 c c 1 c +1 c 1, 45) θ c + 1 c + 1 1 c + 1 c + 1 1, c, +1 c + 1 c +1 1 c+ 1, 46) 1 θ c+ c + 1 1 c + 1 c + 1 1,,1 +1 c + 1 +1 c+1/ 1, 47) θ 1 1,, +1 +1 c+1 1. 3.1. A maximum priciple for the secod order method. I this sectio, we derive a maximum priciple for Scheme I above. The maximum priciple argumet allows for the method to have a time step o oe cell that would violate the CFL coditio o the adacet cell. I particular, we assume 48) ad 49) u t x 1 1+θ u t x +1 1 1+θ, which is the usual CFL coditio for grid blocks I ad I +1. Away from the iterface, where the time step amog adacet cells is the same, the stability of the method is easily demostrated. We summarize the result i the followig propositio. Propositio 3.. Assume a time stepof size t for all elemets to the right of x +1/ ad size t / for all elemets to the left of x +1/, where t satisfies 48) ad 49). The, for Scheme I above with fc) =uc where u>0, with correctios satisfyig 44) 47) o I ad I +1 ad 5) 6) elsewhere, we have 50) sup c sup c 0. Proof. For all blocks except I ad I +1, the boud i sup c follows from the argumets used to prove Propositios.1 ad 3.1. Furthermore, the oly differece betwee the computatio o I ad the stadard method is that there are somewhat differet restrictios o the correctios, amely, they should satisfy 44) 47). Thus, oe ca easily show that c + 1, w + 1, c + 1,, w +1 ad hece are all bouded above by sup c.

HIGHER ORDER LOCAL TIME STEPPING 67 The solutio o I +1 must be more carefully aalyzed. We will first show a maximum priciple for +1. The, this will be used to show maximum priciples for w +1 +1 ad c+1 +1. Cosider +1. The argumet will proceed by the stadard techique of choosig the CFL coditio ad usig the restrictios o the correctios to guaratee that is a covex combiatio of previous values. We defie +1 51) κ 1 =1+ c,1 +1 c c +1 c ad 5) We ca write the computatio of c +1 as 53) +1 = c +1 u t x +1 κ =1+ c, +1 c + 1. c +1 1 c+ [ c +1 + c ) 1 c +1 + c ) )) ] c + 1 + c + 1 [ = c +1 u t c +1 + c,1 +1 x c + c +1 [ = c +1 u t κ 1 c x +1 c ) + κ +1 = c +1 [1 u t x +1 κ 1 + κ ) ) )] + c +1 + c, +1 c + 1 + c + 1 c +1 c + 1 )] ] + u t κ 1 c + u t κ c + 1 x +1 x +1. We ote that the coefficiets of the terms sum to 1. By 44) ad 45), κ i 0, i =1,, ad 54) 1 u t κ 1 + κ ) 0 x +1 if the time step is chose such that 49) is satisfied. By the above-stated boud o c + 1, we ca take the absolute value of each side of 53) ad obtai 55) ] +1 = c +1 [1 u t κ 1 + κ ) + u t κ 1 c + u t κ c + 1 x +1 x +1 x +1 sup c. Next, cosider w +1 +1. We defie two ew terms: 56) ad 57) κ 3 =1+, +1 c + 1 +1 c+ 1 κ 4 =1+,1 +1 +1. c+1

68 CLINT DAWSON AND ROBERT KIRBY The method ca be writte as 58) w +1 +1 = c+1 +1 u t x +1 [ = +1 u t +1 x +,1 +1 +1 = +1 u t [ x +1 = +1 [ +1 + +1 1 ) ] c + 1 + c + 1 + + c + 1 + c + 1 κ 3 ) + +1 +, +1 1 c+ [1 u t x +1 κ 3 + κ 4 ) ] + u t +1 )] + )] ) + κ 4 +1 c+1 x +1 κ 3 c + 1 + u t x +1 κ 4. As before, the coefficiets sum to 1. By a similar argumet, 46) ad 47) give the oegativity of each κ term, ad we have the same CFL restrictio 49) as above. Takig the absolute value of each side of 58), we see that ) w +1 +1 max +1, 1 c+,, ad each of these terms is bouded by the solutio at time. Fially +1 = 1 c +1 + w +1 ) +1 1 c +1 + 1 w+1 +1 sup c. To summarize, our method is coservative the flux used at the right edge of elemet I is equal to that used at the left edge of I +1 ). We have demostrated stability subect oly to local CFL restrictios. Next we discuss the limitig of the correctios so that the coditios 44) 47) are satisfied. Later, we will preset umerical evidece demostratig that local time steppig does ot appear to degrade the accuracy of the method. 3.. Computigad limitigthe correctios. We discuss i slightly more detail the computatio of the c terms. We assume as before that θ is chose so that 0 θ 1. First, assume c, c 1 15). We the set, ad c,1 +1 are computed so that 44) is satisfied, say, usig 59) c, { c,1 +1 = +1 if c,1 +1 θ c +1 c+1/, 0 otherwise. I a similar way, correctios / ad / 1 are computed so that the first iequality i 45) is satisfied, ad / is limited so that 60) Hece, oe ca show 61) / θ c +1 /. 0 κ 1,κ 1+θ.

Proceedig, / is satisfied, ad so that HIGHER ORDER LOCAL TIME STEPPING 69 ad / 1 are computed so that the first iequality i 46) 6) / θ +1 c+1/ ;,1 +1 is computed so that it also satisfies 63),1 +1 θ +1 c+1/. Similarly, 1,, ad, +1 are computed so that 47) is satisfied. Thus 64) 0 κ 3,κ 4 1+θ. 3.3. The case u<0. Cosider Figure agai, ow with fc) =uc ad u<0. This case is aalogous to the situatio i which u>0 ad the picture i Figure is reversed, amely, the local time steps are to the right of the iterface. I this case, to compute the solutio i elemets I ad I +1, we use the followig procedure. Scheme II u <0). i) Compute correctios c, c,1 +1, ad c + so that 65) is satisfied. ii) c + 1 = c uλ [c +1 c,1 +1 c c )]. iii) Compute / ad c, +1 so that 66) is satisfied. iv) w + 1 = c + 1 uλ [c +1 c, +1 1 c+ c + 1 )]. v) c + 1 = 1 c + 1 w+ ). vi) Defie c +1 1 c,1 +1 + c, +1 ). vii) +1 = c +1 [ uλ +1 c + c + c +1 )] c +1. viii) Compute c + 1,,1 +1, ad + so that 67) is satisfied. ix) = / x) Compute uλ [c+1 +1,1 +1 c + 1 c + 1 )]. ad, +1 to satisfy 68). xi) w +1 = uλ [c+1 +1, +1 )]. xii) = 1 1 c+ + w +1 ). xiii) Defie +1 1 +1,1 c +1 +, +1 ). xiv) w +1 +1 = c +1 uλ +1 [c+1 + + +1 )]. xv) +1 = 1 c +1 + w +1 ) +1. The coditios o the correctios i this case are 65) 66) θ c,1 +1 c c +1 c, c + c,1 +1 c + 1, c +1 θ c, +1 / c +1, c + c, +1 c+1/ c + 1, c +1

70 CLINT DAWSON AND ROBERT KIRBY 67) 68),1 +1 / θ +1, c+1/ +1 c +, +1 θ +, c+1 +1 c +1 +,1 +1 + c+1 +1 +1, c +1 +1 c+1 These coditios lead to the followig result. The proof is left to the reader. Propositio 3.3. Assume a time stepof size t for all elemets to the right of x +1/ ad size t / for all elemets to the left of x +1/, where t satisfies 48) ad 49) with u replaced by u). The, for Scheme II above with fc) =uc where u< 0, with correctios satisfyig 65) 68) o I ad I +1 ad 5) 6) elsewhere, we have 1, 1. 69) sup c sup c 0. 3.4. Extesio to a oliear flux fuctio. The extesio of the methods above to a more geeral umerical flux h essetially combies the ideas of Schemes I ad II above. I particular, the method is as follows. Scheme III geeral flux fuctio). i) Compute correctios c 1, c, c,1 +1, ad c + so that 44) ad 65) are satisfied. ii) c + 1 = c λ [hc + c,c +1 c,1 +1 ) hc 1 + c 1,c c )]. iii) Compute / 1, /, ad c, so that 45) ad 66) are satisfied. iv) Compute w + 1 w + 1 = c + 1 λ by v) c + 1 = 1 c + 1 w+ ). vi) Defie c +1 1 c,1 +1 + c, vii) Compute +1 [ ] hc + 1 + c + 1,c +1 c, +1 ) 1 hc+ 1 + c + 1 1, c + 1 c + 1 ). +1 by +1 = c +1 λ +1 +1 ). [ hc +1 + c +1,c + c +) 1 ] hc + c,c +1 c,1 1 +1 )+hc+ + c + 1,c +1 c, +1 )). viii) Compute c + 1 1 1, c+,,1 +1, ad + so that 46) ad 67) are satisfied. ix) Compute by = c + 1 λ [ ] hc + 1 + c + 1, +1,1 +1 ) hc + 1 1 1 1 + c+ 1,c+ c + 1 ). x) Compute 1,, ad, +1 so that 47) ad 68) are satisfied. xi) Compute w +1 by w +1 = λ xii) = 1 c+ 1 + w +1 ). [ ] h +, +1, +1 ) h 1 + 1, ).

xiii) Defie +1 1,1 c +1 + c, +1). xiv) w +1 +1 = c+1 +1 t HIGHER ORDER LOCAL TIME STEPPING 71 x +1 [ h +1 + +1, + + ) 1 hc + 1 + c + 1, +1,1 +1 )+h +, +1, +1 )) ]. xv) +1 = 1 c +1 + w +1 ) +1. 3.5. Extesio to M steps. Now suppose we take M steps i grid block I to oe step i block I +1. The easier extesio of the method described above is for the case where M is eve, although a extesio to M odd ca also be made. Assume M is eve. Whe M =, ote that to compute +1 we used the values c up through c+1/. The same is true i geeral. We compute / by takig M/ steps i block I. The average of the fluxes alog edge +1/ over these steps is used to compute +1, ust as above. The computatio of w+1 +1 is aalogous. We take M/ steps to compute +1, startig with c+1/, ad the average of the fluxes alog edge +1/ is used to compute w +1 +1. The maximum priciple argumet ca be carried through with appropriate limitig of the correctios. Similar to the case M =, at each substep i, i =1,...,M/, we compute a correctio c,i +1 i block I +1 so that the aalogues of 44), 45), 65), ad 66) are satisfied. The fial correctio c +1 is the average of all of these substep correctios. Similarly, oce +1 is computed, correctios,i +1 are computed at each substep so that the aalogues of 46), 47), 67), ad 68) are satisfied, ad +1 is computed by averagig these correctios. 4. Numerical results. Here we preset results examiig the accuracy ad stability of the method i sectio 3. 4.1. Liear example: Smooth problem. We will first show how local time steppig affects the errors ad stability i the case of a smooth liear problem. Cosider iitial ad boudary coditios such that the true solutio is siπx t)). The space-time domai is [0, 1]. We have the followig cases: 1. Uiform mesh of width x, global time step t = x 3.. Regio [0, 0.5] refied to x x, time step globally refied to t = 3. 3. Regio [0, 0.5] refied to x, time step o [0, 0.5] reduced locally) by a factor of. Examiig Tables 1,, ad 3, we see that the rate of covergece i L i all three cases is aroud 1.5, which is to be expected for a mimod-limited method. Coceivably, the local time steppig could icur a Oh 1 ) error which a costat time step would ot icur. Thus, we tured off the limiter o the slopes ad saw that the local steppig does ot degeerate the order of accuracy, as we see secod order covergece i Tables 4 ad 5. 4.. Liear example: Rough problem. Now that we have see that the local time steppig scheme does ot degeerate the accuracy of the approximatio, we will show that the method is ideed stable with a local CFL coditio. Cosider the liear model case of f c) =c with cx, 0) = 1 for x<0 ad cx, 0) = 0 for x>0. At t =0.5, the true solutio is a frot at x =0.5. Cosider the followig situatios:

7 CLINT DAWSON AND ROBERT KIRBY Table 1 Case 1: Uiform mesh, global time step. x L error L 1 error.500d+00.1447d+00.105d+00.150d+00.4995d-01.387d-01.650d-01.181d-01.14d-01.315d-01.6033d-0.448d-0.156d-01.1955d-0.187d-0.781d-0.691d-03.3714d-03 rate: 1.56 1.66 Table Case : Local refiemet, global time step. x L error L 1 error.500d+00.1145d+00.944d-01.150d+00.4116d-01.646d-01.650d-01.1461d-01.8774d-0.315d-01.4869d-0.550d-0.156d-01.1586d-0.736d-03.781d-0.511d-03.1965d-03 rate: 1.56 1.77 Table 3 Case 3: Local refiemet, local time step. x L error L 1 error.500d+00.1306d+00.1030d+00.150d+00.4304d-01.641d-01.650d-01.160d-01.1018d-01.315d-01.5981d-0.338d-0.156d-01.18d-0.105d-0.781d-0.7441d-03.3196d-03 rate: 1.48 1.63 Table 4 Uiform mesh ad time step o [0, 1], ulimited slopes. x L error L 1 error.500d+00.1380d+00.1317d+00.150d+00.4331d-01.3371d-01.650d-01.1147d-01.9483d-0.315d-01.869d-0.417d-0.156d-01.7079d-03.6061d-03.781d-0.1746d-03.1509d-03 rate: 1.94 1.95 Table 5 Uiform mesh ad local time step o [0, 1], ulimited slopes. x L error L 1 error.500d+00.190d+00.1113d+00.150d+00.3908d-01.951d-01.650d-01.9640d-0.784d-0.315d-01.35d-0.1785d-0.156d-01.594d-03.4355d-03.781d-0.181d-03.1078d-03 rate:.0.01

HIGHER ORDER LOCAL TIME STEPPING 73 1 Global time steppig, CFL umber = /3 c 0.8 0.6 0.4 0. 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 3. Global time steppig, model advective frot. 8e+4 Istability icurred by mesh refiemet 6e+4 4e+4 e+4 0 -e+4-4e+4-6e+4-8e+4 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 4. Local mesh refiemet creates istability. 1. Global uiform mesh, x = 1 18, global CFL umber of 3.. Refie the mesh o [0, 0.5] by a factor of, keep the same global time step. CFL umber i first half of the domai is 4 3.) 3. Refie the time step o [0, 0.5] by a factor of, givig a local CFL umber of 3 everywhere. Observe that the frot is propagated stably i the first case i Figure 3 but whe the CFL coditio is violated i the secod case, we icur massive istability Figure 4). However, the local time steppig method, with its local CFL coditio, gives a stable ad accurate approximatio to the frot i Figure 5.

74 CLINT DAWSON AND ROBERT KIRBY 1 Global time steppig, CFL umber = /3 c 0.8 0.6 0.4 0. 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 5. Local time steppig restores stability. 4.3. Noliear example: Buckley Leverett. Next, we examie a case where the flux is oliear. The Buckley Leverett problem, give by 70) f c) = c c + a 1 c), is a stadard test problem arisig i two-phase flow i porous media. This flux fuctio is Lipschitz but ocovex. I the case of a = 0.5, it is kow from the Rakie Hugoiot coditio that the frot is a rarefactio dow to the poit where c =0.44, ad the the solutio umps to c = 0. See, for example, [10] for a discussio. The Rakie Hugoiot coditio also gives that this frot propagates at a velocity of 1.6. We performed a series of umerical experimets o a uiform mesh. First, we used global time steppig to verify that the code put the shocki the right locatio with the correct ump. The, we refied the time step i the first half of the domai repeatedly i order to verify that the local time steppig did ot alter the method s shock-capturig abilities. Each of these cases was ru with a mesh spacig of x = 1 1 18 ad a mai time step of t = 160 to time t =0.5. The true frot should be at x =0.81. These frots were all ear, though diffused, ad further spatial refiemets gave sharper frots at the right spot. Figure 6 shows these results. Further, we show that we have stability oly subect to a local CFL costrait. Agai, we cosider three cases, with x max = 1 18 ad t max = 1 160. I each case, the mesh width is 1 x max o [0, 0.5] ad x max elsewhere. 1. Global time step t max.. Global time step 1 t max. 3. Local time steppig. As see i Figure 7, the large time step i the presece of the local refiemet has caused some mild istability with over- ad udershoot. Due to the large velocity, this mild oscillatio has propagated out of the first part of the domai ad ito the secod. However, whe the time step is refied either globally or locally i the presece of the mesh refiemet, the frot is well approximated.

HIGHER ORDER LOCAL TIME STEPPING 75 1 0.9 Coservatio properties o Buckley-Leverett frot gts -1 local time-steppig 8-1 local time-steppig 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 6. Local time steppig approximatios to Buckley Leverett frot at x =0.81. 1 0.9 Stability properties for Buckley-Leverett frot Case 1 Case Case 3 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 7. Local time steppig cures istability caused by local mesh refiemet.

76 CLINT DAWSON AND ROBERT KIRBY 1 0.9 0.8 0.7 0.6 Y 0.5 0.4 0.3 0. 0.1 C 0.9 0.7 0.57149 0.48571 0.85714 0.14857 0-0.14857-0.85714-0.48571-0.57149-0.7-0.9 0 0 0.5 1 X Fig. 8. Iitial coditio for Burgers equatio, cosistig of a coe of height 1 i the lower left corer, ad a coe of height 1 i the upper right corer. 5. Two-dimesioal Burgers equatio. The local time steppig schemes described above have bee exteded to two space dimesios. The spatial discretizatio we have employed is based o a high resolutio method developed by Durlofsky, Egquist, ad Osher [4] for ustructured, triagular grids. The differece betwee what we have implemeted ad the method i [4] is i the slope-limitig step. Three differet slopes are costructed usig liear iterpolatio with the elemet ad its eighbors. The the steepest slope which does ot itroduce overshoot/udershoot at the edge boudaries is selected. The two methods differ i the case where o slopes satisfy this coditio. I our implemetatio, the slope is set to zero i this case, whereas Durlofsky, Egquist, ad Osher simply choose the iterpolat with the smallest gradiet. Some results for local time steppig applied to the trasport equatio 1), with highly varyig velocity fields, ca be foud i [9]. Here, we apply local time steppig to the two-dimesioal iviscid Burgers equatio 71) c t + f c) x + f c) y =0, where f c) = c. We take as a iitial coditio a fuctio cosistig of two coe shapes, oe with height 1 ad the other with height 1. The iitial coditio is displayed i Figure 8. I Burgers equatio, larger cocetratios give rise to larger velocities. Cosequetly, the ceters of the coes are advected alog at a higher rate tha the edges. Moreover, the positive coe has positive velocity ad the egative coe egative velocity; thus, the coes approach each other ad collide i the ceter of the domai. This geeral behavior is give i Figure 9, where we have show the solutio at time

HIGHER ORDER LOCAL TIME STEPPING 77 1 0.9 0.8 0.7 0.6 Y 0.5 0.4 0.3 0. C 0.57149 0.48571 0.85714 0.14857 0.001-0.001-0.14857-0.85714-0.48571-0.57149 0.1 0 0 0.5 1 X Fig. 9. Global time steppig solutio at T =1.1. T =1.1. This solutio was obtaied usig the method i [4], modified as metioed above, with Heu s method used for the temporal itegratio. The fiite elemet mesh for this case, show i Figure 10, cosists of 15489 triagles ad is ustructured. A extesio of the secod order local time steppig scheme described previously was implemeted for this problem. I our implemetatio, the elemets take either the global CFL time step or a time step M times larger, assumig that this larger time step does ot violate the local CFL costrait. The time steps are redistributed throughout the domai after each large time step. Numerical results for M =,5, ad 10 were geerated. Relative L 1 ad L errors betwee the local ad global time steppig solutios were computed at time T =1.1. The relative L 1 error is E Error L 1 = c globalx E ) c local x E ) me) 7) E c, globalx E ) me) where the sum o E is over all elemets, x E is the baryceter of elemet E, c global is the global time steppig solutio, c local is the local time steppig solutio, ad me) is the area of triagle E. A similar defiitio holds for the relative L error. These errors are give i Table 6. Note that they are o the order of 1%. Table 7 shows the CPU ru times i secods for each of these cases. Notice a speedup of about 1.7.4 for the M:1 time steppig scheme over the global time steppig scheme. For this problem, most elemets ear the peaks of the coes take the smaller global CFL) time step, as these elemets have larger velocity. As the simulatio proceeds, the solutio spreads ad evetually more elemets are forced to take the smaller time step. This pheomeo is see i Figures 11 ad 1, where the local time step distributio is plotted at times T =.1 ad T =1.1, respectively, for the case M = 5. For the M = 10 case, the small time step regio is more extesive, because fewer elemets meet the criterio ecessary to take the larger time step. Thus, ay gais i icreasig M are offset somewhat, ad cosequetly there

78 CLINT DAWSON AND ROBERT KIRBY 1 0.9 0.8 0.7 0.6 Y 0.5 0.4 0.3 0. 0.1 0 0 0.5 1 X Fig. 10. Fiite elemet mesh for Burgers equatio test case. Table 6 Relative L 1 ad L errors for M:1 time steppig schemes. M Rel. L 1 error Rel. L error.0041.0130 5.0044.0086 10.0040.0074 Table 7 Trasport ru times i secods for M:1 time steppig schemes. M Ru time sec) 1 533 1480 5 1116 10 1075 is ot a dramatic decrease i ru time as we icrease M for this problem. I Figure 13, we have plotted the percetage of elemets takig the global CFL time step vs. simulatio time for M =, 5, ad 10. Here we see that for M =, iitially about 3.5% of the elemets take the global CFL time step, ad the percetage grows to about 7% by the fial time, T =1.1. For M = 5, the rage is from about 0% to 7%, ad for M = 10, the rage is about 8% to 37%. For M too large, of course, all elemets would be forced to take the global CFL time step.

HIGHER ORDER LOCAL TIME STEPPING 79 1 0.9 0.8 0.7 0.6 Y 0.5 0.4 0.3 0. 0.1 0 0 0.5 1 X LTS 4.75 4.5 4.5 4 3.75 3.5 3.5 3.75.5.5 1.75 1.5 1.5 Fig. 11. Distributio of local time steps for M =5ad T =.1. Lighter regio idicates where smaller time steps were take. 1 0.9 0.8 0.7 0.6 Y 0.5 0.4 0.3 0. 0.1 0 0 0.5 1 X LTS 4.75 4.5 4.5 4 3.75 3.5 3.5 3.75.5.5 1.75 1.5 1.5 Fig. 1. Distributio of local time steps for M =5ad T =1.1. Lighter regio idicates where smaller time steps were take.

80 CLINT DAWSON AND ROBERT KIRBY 40 35 30 5 percetage 0 15 10 5 0 0 0. 0.4 0.6 0.8 1 1. 1.4 time Fig. 13. Percetage of elemets takig the global CFL time step as a fuctio of simulatio time for M =solid lie), M = 5 +), ad M =10 ). 6. Coclusios. We have developed ad proved maximum priciples for local time steppig schemes based o first ad secod order time discretizatios, ad piecewise liear spatial discretizatios. Numerical results give here ad i [9] idicate that these local time steppig schemes exhibit similar accuracy ad stability to the global time steppig schemes upo which they are based, at a fractio of the computatioal cost. I this paper, we have ot addressed the issue of whether our local time steppig schemes satisfy additioal properties TVB, etc.) from which we could coclude that the schemes coverge to the etropy solutio. However, all of our umerical results to date idicate that the local time steppig solutios are almost idetical to those obtaied by global time steppig. Thus, our experieces so far lead us to believe that our local time steppig schemes iherit the covergece properties of the related global scheme, though this has ot bee prove. REFERENCES [1] M. Berger ad J. Oliger, Adaptive mesh refiemet for hyperbolic partial differetial equatios, J. Comput. Phys., 53 1984), pp. 484 51. [] B. Cockbur ad C.-W. Shu, TVB Ruge-Kutta local proectio discotiuous Galerki fiite elemet method for scalar coservatio laws II: Geeral framework, Math. Comp., 5 1989), pp. 411 435. [3] C. Dawso, High resolutio upwid-mixed fiite elemet methods for advectio-diffusio equatios with variable time-steppig, Numer. Methods Partial Differetial Equatios, 11 1995), pp. 55 538. [4] L. Durlofsky, B. Egquist, ad S. Osher, Triagle based adaptive stecils for the solutio of hyperbolic coservatio laws, J. Comput. Phys., 98 199), pp. 64 73. [5] J. E. Flaherty, R. M. Loy, M. S. Shephard, B. K. Szymaski, J. D. Teresco, ad L. H. Ziatz, Adaptive local refiemet with octree load-balacig for the parallel solutio of three-dimesioal coservatio laws, Joural of Parallel ad Distributed Computig, 47 1997), pp. 139 15.

HIGHER ORDER LOCAL TIME STEPPING 81 [6] S. Gottlieb ad C.-W. Shu, Total variatio dimiishig Ruge-Kutta schemes, Math. Comp., 67 1998), pp. 73 85. [7] A. Harte ad S. Osher, Uiformly high-order accurate ooscillatory schemes. I, SIAM J. Numer. Aal., 4 1987), pp. 79 309. [8] J. Kallideris ad J. Baro, Adaptatio methods for viscous flows, i Computatioal Methods i Viscous Aerodyamics, C. A. Brebbia, ed., Elsevier, Amsterdam, 1990. [9] R. Kirby, A Posteriori Error Estimates ad Local Time-Steppig for Flow ad Trasport Problems i Porous Media, Ph.D. thesis, Uiversity of Texas at Austi, Austi, TX, 000. [10] R. Leveque, Numerical Methods for Coservatio Laws, Birkhäuser-Verlag, Bosto, 1990. [11] S. Osher ad R. Saders, Numerical approximatios to oliear coservatio laws with locally varyig time ad space grids, Math. Comp., 41 1983), pp. 31 336. [1] C.-W. Shu, TVB uiformly high-order schemes for coservatio laws, Math. Comp., 49 1987), pp. 105 11. [13] C.-W. Shu, Total-variatio-dimiishig time discretizatios, SIAM J. Sci. Statist. Comput., 9 1988), pp. 1073 1084. [14] B. va Leer, Towards the ultimate coservative differece scheme, V. A secod order sequel to Goduov s method, J. Comput. Phys., 3 1979), pp. 101 136.