MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES BASED ON SLOPE LIMITERS

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1 MATHEMATICS OF COMPUTATION S Article electronically published on May 6, 0 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES BASED ON SLOPE LIMITERS ORHAN MEHMETOGLU AND BOJAN POPOV Abstract. A maximum principle and convergence of second order central schemes is proven for scalar conservation laws in dimension one. It is well known that to establish a maximum principle a nonlinear piecewise linear reconstruction is needed and a typical choice is the minmod limiter. Unfortunately, this implies that the scheme uses a first order reconstruction at local extrema. The novelty here is that we allow local nonlinear reconstructions which do not reduce to first order at local extrema and still prove maximum principle and convergence.. Introduction This paper is concerned with scalar hyperbolic conservation laws { ut + fu. x =0, x, t R 0,, ux, 0 = u 0 x, x R, where f is a given flux function. In recent years there has been a lot of development in mathematical theory and construction of numerical algorithms for.. Even though the existence-uniqueness theory is complete, there are many numerically efficient methods for which the questions of convergence and error estimates are still open. For example, there are many second or higher order nonoscillatory schemes based on minmod or other limiters which are numerically robust but theoretical resultsaboutconvergenceorerrorestimatesarestillmissing[,4,5,6,7,8]. A fundamental step in the design of such a numerical algorithm is a piecewise linear slope reconstruction; see for example [4, 8]. Nonlinear limiters such as minmod, generalized minmod, UNO, and superbee are widely used. A common shortcoming of all proofs is that the linear reconstruction used must reduce to first order zero slope at local extrema in order to prove a maximum principle for the scheme see for example [8], and when that limitation is not imposed there are typically no theoretical results for the methods even though they perform well in practice [,4,8,]. In this paper we consider a class of nonlinear reconstructions which include and are motivated by the so-called minimum angle plane reconstruction MAPR introduced in []. The key idea see section here or [], is that at local extrema the Received by the editor May, 00 and, in revised form, September 9, 00 and December, Mathematics Subect Classification. Primary 65M; Secondary 65M08. This material is based on work supported by the National Science Foundation grant DMS This publication is based on work partially supported by Award No. KUS-C-06-04, made by King Abdullah University of Science and Technology KAUST. c 0 American Mathematical Society Reverts to public domain 8 years from publication

2 ORHAN MEHMETOGLU AND BOJAN POPOV slope of the reconstruction is not set to zero but it is limited by the smallest local slope. As a consequence, the maximum principle proofs for generalized minmod limiters in [8, 6] are no longer valid and a new approach is needed. The paper is organized as follows. In section, we give a short overview of the Nessyahu-Tadmor NT staggered central scheme and prove a maximum principle when MAPR-like limiters are used. The result holds for a conservation law with a Lipschitz flux see Theorem., and also for any smooth convex flux see Theorem.5. In section, we prove that in the case of strictly convex flux the NT scheme satisfies a one-sided Lipschitz stability estimate and converges to the entropy solution when the MAPR-like limiter is properly selected. This is a generalization of the result in [0] for the minmod slope reconstruction.. Nonoscillatory central schemes In this section we are going to consider second order nonoscillatory staggered central difference approximations to the scalar conservation law,. u t + fu x =0. We restrict our attention to the one-dimensional staggered NT scheme [8], which was the motivation for the construction of many other central staggered schemes; see for example [6, 7,, ]. Even though it may be possible to prove more general results, we will only consider the standard NT scheme setup. Let vx, t bean approximate solution to., and assume that the space mesh Δx and the time mesh Δt are uniform. Let x := Δx, Z, I := {s : s x < Δx }, tn = nδt, n N, and. v n := vx, t n dx Δx I be the average of vx, t n overi. Next, assume that v,t n is a piecewise linear function of the form. vx, t n = L x, t n := v n +x x Δx v χ x, where χ is the characteristic function over I and Δx v is the numerical derivative of vx = x,t n, which is yet to be specified. We proceed by integrating. over I + tn,t n+ which yields v + tn+ = x+ x+ L x, t n dx + L + x, t n dx Δx x x +.4 t n+ t n+ fvx +,τdτ fvx,τdτ. Δx t n t n The first two linear integrands on the right of.4, L x, t n andl + x, t n, can be integrated exactly and if the CFL condition with mesh ratio λ := Δt Δx.5 λ max x x x + f vx, t, Z, is satisfied, then the last two integrands on the right of.4, fvx,τ and fvx +,τ, can be integrated approximately by the midpoint rule at the expense

3 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES of the OΔt local truncation error. Thus we arrive at.6 v n+ = + vn + v++ n 8 v v + λ fv n+ + fvn+. The approximate value for the midpoint rule in time, v n+, satisfying second order accuracy requirement can be chosen as.7 v n+ = v n λf, owing to Taylor expansion and.. Here, Δx f stands for an approximate numerical derivative of the flux fvx = x,t n, which is yet to be determined. Although there are many different recipes to construct v and f, in this paper the following are considered to be the approximations of the numerical derivatives.8 v =mv+ n v n,v n v, n.9 f = f v n v, where m, is the modified minmod limiter {.0 ma, b := sga min a, b, ab 0, σ min a, b, ab < 0, with σ R, σ. Remark.. Note that the choice. σ =sgs, where s = recovers the MAPR limiter introduced in []. { a, a b, b, b a, Using the approximate slopes.8 and the approximate flux derivatives.9, we end up with a family of central differencing schemes in the predictor-corrector form.. v n+ = v n λf, v n+ = + vn + v++ n 8 v v + λ fv n+ + fvn+. Note that.0 allows the predicted values {v n+ } see. to violate maximum principle. That is, the minimum/maximum of the sequence {v n+ } could be smaller/larger than that of {v n}. This is going to be the main difficulty in proving maximum principle. We begin with a result in a simpler setting when the flux is globally Lipschitz continuous. Theorem.. Let v be chosen by.8 and f = f v nv. If f is globally Lipschitz continuous, then the scheme described by.. under the CFL condition.4 λ f L R β satisfies the maximum principle.5 minv n,v+ n v n+ maxv + n,v+. n

4 4 ORHAN MEHMETOGLU AND BOJAN POPOV Proof. First, we rewrite the term fv n+ + fvn+ in. as.6 fv n+ + fvn+ =f ξ n+ + v n+ + vn+, where minv n+,v n+ + ξn+ maxv n+ +,v n+ +. Observe that, v n+ + vn+ = vn + v n λ f v+v n + f v n v.7 + λ f v n + + f v n v+ n v n + β v+ n v n. Using.4,.6 and.7 we find an upper bound for., v n+ + vn + v++ n 8 v v + + λ fv n+ + fvn+.8 vn + v++ n + β v + v vn + v n ++ v + v =maxv n,v n +, and similarly a lower bound, v n+ + vn + v+ n 8 v v + λ fv n+ + fvn+.9 vn + v+ n + β v + v vn + v n + v + v = minv n,v n +. The above two bounds prove the theorem. Remark.. This maximum principle proof implies the TVD bound for the NT scheme proven in Theorem. from [8]. The proof of the TVD bound in [8] is not valid for the Jacobian form which we use here see.8.9 in this paper or.6a.6b in [8], even for the standard minmod limiter σ = 0 in.0. Hence, the present result extends Theorem. from [8] for the special choice of numerical derivative and numerical flux.8.9 considered here. Definition.4. The range of a function g : R R is defined to be the interval Rg := [essinf x R gx, esssup x R gx]. Theorem.5. Let v be chosen as in.8 and f = f v nv. If f is strictly convex, that is, there exists constants γ γ such that.0 0 <γ f γ, then the scheme described by.,. satisfies the maximum principle.5 under the CFL condition. λ max f w β, w Ru 0 where β is a fixed constant which depends only on γ and γ ;see..

5 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES 5 Proof. First, we observe b b t fb fa = f tdt a = a + b f sds + f dt a+b a. γ b a + a + b 4 f b a. Let a = v n+ and b = v n+ + in the above inequality, then fv n+ + fvn+ γ. v n+ + 4 vn+ + f vn+ + v n+ + n+ v + vn+ Next, we bound the terms appearing on the right-hand side. We start with v n+ + vn+ = vn + v n λ f v+v n + f v n v + λ f v n + + f v n v+ n v n.4 + β v+ n vn +β f v+ n γ f v n + β γ max f v n. Note that the inequality.5 λ f v n v + f v n 4 +v + β vn + v n implies.6 minv n,v+ n vn+ + v n+ + maxv n,v n + for all β. From.0 and.6 it follows that.7 f vn+ + + vn+ max f v n. We use.,.4 and.7 and derive fv n+ + fvn+ γ + β + max f v n γ.8 γ + β + γ v n+ + vn+. + βmax f v n v+ n v n. Using.8 in. gives the estimates v n+ + vn + v++ n 8 v v + + λ fv n+ + fvn+.9 vn + v++ n γ + β + β + β + δ n 4 γ +

6 6 ORHAN MEHMETOGLU AND BOJAN POPOV and.0 v n+ + vn + v+ n 8 v v + λ fv n+ + fvn+ vn + v+ n γ + β + β + β + 4 γ where δ n = v + + n vn. Hence, under the CFL condition γ + β. β + β + γ 4, we have. minv n,v+ n v n+ maxv + n,v+. n δ n +, Remark.6. In the case of Burgers equation. holds when β 0.97 which implies that the CFL condition. is satisfied with β =0.97. Remark.7. It is possible to prove a result similar to Theorem.5 for nonconvex fluxes which satisfy. 0 <γ f k x γ. Such functions are called k-monotone functions and any fixed polynomial is a k- monotone function for some k. The proof of this result will be given elsewhere.. One-sided l stability and convergence Recall that {v n} Z and {v n+ } + Z are the sequences of cell averages of the numerical solution of NT scheme at time t n and t n+, respectively. Let us introduce the following notation for the ump sequences of the NT solution. δ + := vn + v n and δ := v n+ + v n+, Z, at times t n and t n+, respectively. With this notation we have the following theorem which is the main result of this section. Theorem.. Let u 0 L R, letf be strictly convex in Ru 0 and let f be bounded on R. That is, there exist constants γ and γ such that.. 0 <γ f w, w Ru o, f x γ, x R. Then there exists a constant β which depends only on the ratio γ /γ such that under the CFL condition λ max f w β, w Ru 0 the NT scheme with the limiter.0 and 0 σ satisfies the following onesided Lipschitz condition.4 {δ + } Z l {δ + +} Z l, whereweusethestandard + notation: x + =maxx, 0. In other words, the l norm of the positive umps does not increase in time.

7 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES 7 Remark.. Taking σ = 0 in.0, we obtain the original minmod limiter. It is easy to see that among all σ, σ, the choice σ = in.0 minimizes the size of the positive umps in the piecewise linear numerical solution. These are the so-called entropy violating umps for convex flux see [9] and Remark on page 4 in [8], and one needs to have control of their size in order to prove convergence to the entropy solution. Proof. It is an easy exercise in real analysis to show that every bounded sequence can be decomposed into a union of monotone subsequences. That is, given the sequence {v n} Z there exists a nonempty collection of index sets Λ k :={ min k max} k such that max k = k+ min for all k Z, and{vn } Λ k is nondecreasing if k is even, and nonincreasing if k is odd. This decomposition is not necessarily unique. To fix one, we choose {Λ k } k such that Λ k has maximum number of terms for each even k. That is, for all even k such that Λ k is nonempty, we have.5.6 v k min > v k min, v k max + > v k max. Note that we have a single set Λ k if the data is monotone and that.5 and.6 only make sense if min k > and k max <, respectively. With this notation, there are only two possibilities to generate nonnegative umps δ in the new sequence {v n+ +/ } Z starting from the old sequence {v n} Z: i If v n vn vn +, i.e.,,+ Λ k for some even k, thenwehavean internal ump, i.e., generated from the interior of a nondecreasing monotone subsequence Λ k. ii If v n vn vn + vn 0 and at least one of these umps δ or δ + is not zero. That is,, Λ k and, + Λ k+ for some k, thenwe have a boundary ump, i.e., generated on the boundary of Λ k and Λ k+. The umps generated in case are always nonnegative, whereas the umps generated in case may have different signs. Next, without loss of generality, we assume that there is at least one nondecreasing subsequence of {v n} Z, say with index set Λ 0 = {0,...,m}. Otherwise,there is nothing to prove. We define the modified cell averages { v n} Z, as v0 n, 0,.7 v n := v n, 0 <<m, vm, n m, which is the so-called constant extension of {v n}m =0 ; see [0]. The umps of the modified cell averages are given by.8 δ+ := vn + vn and δ := v n+ + The following facts follow from the definition of { v n } Z: v n+, Z..9 v n+ = v + 0 n,, v n+ = v + m, n m, v + = v +, m.

8 8 ORHAN MEHMETOGLU AND BOJAN POPOV Using the above we get δ =...,0, δ 0, δ,δ,...,δ m, δ m, δ m, 0,...,.0 δ + =...,0,δ,δ,...,δ m, 0,..., with the convention that we drop any terms that do not make sense when m. In view of i and ii the nonnegative umps of the new sequence can be decomposed into interior and boundary umps generated by each Λ k, for even k. Therefore, to prove the theorem it is enough to show that. {δ + } m =0 l {δ + +} m =0 l, because the left-hand side includes all nonnegative umps that may be generated by Λ 0. For the sequence.7 the limiter.0 coincides with the minmod limiter. Hence, we can apply the one-sided stability result from [0] for a single nondecreasing sequence m. δ δ +. =0 =0 =0 Having. it suffices to show that. δ + δ. =0 We split the proof of. into four cases. Case. m 4. We need the following lemma. Lemma.. The following inequality holds when σ 0 in.0:.4 δ + +δ 0 + δ + δ 0. Proof. We are going to prove the argument in two steps. Step. First, we will show that δ δ 0. Observe that.5 δ δ =δ + δ δ δ =k n+ v n+ = k n+ v n+ v n+ v n+, v n+ v n+ where k n+ := δ + δ > 0asδ and δ are both positive. Next, we need to check the sign of v n+ v n+,where.6.7 v n+ v n+ = vn 0 + v n + 8 v 0 v λ = vn 0 + v n + 8 v 0 v λ f v n+ f v n+ 0 fv n+ fv n+ 0 Note that v =mv n v n 0,v n v n = v,andv n+ = v n λ f v n v = v n+, which follows from.9. Also observe that v 0 =m0,v n v n 0 =0andv 0 0,.

9 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES 9 since σ 0 in.0. Subtract.7 from.6 to get.8 v n+ v n+ = 8 v 0 v 0+λ = 8 v 0 + λ f v n+ 0 fv n+ fv n 0 fv n+ 0 8 v 0 + λ f v n 0 v 0 max 0 f v0 n, f v0 n λ f v0 n v 0. There are two possibilities: I Suppose max f v0 n, f v0 n λ f vo n v 0 = f v0 n. Then we rewrite.8 as.9 v n+ v n+ 8 v 0 + λf v0 n v 0 8 β + v 0 0. II Suppose max f v n 0, f v n 0 λ f v n 0 v 0 = f v n 0 λ f v n 0 v 0. Then by the mean value theorem and. we have.0 v n v n 0 γ max f v n, f v n 0... Now, by the above inequality, Taylor expansion, and. we have f v0 n λ f v0 n v 0 f v0 n + λ γ v 0 f v0 n +λ γ max f v n, f v0 n γ f v0 n +β γ. γ We use this result in.8 to get v n+ v n+ 8 v 0 + λ v f 0 n λ f v0 n v 0 +β γ 8 v 0 + λ 8 β + γ +β γ γ Therefore, in both cases I and II, we conclude that. v n+ v n+ 0. f v n 0 v 0 v 0 0. f v n 0 v 0 We finish the proof of Step by observing that.5 and. imply δ δ 0. Step. We now prove that δ 0 + δ 0 0.

10 0 ORHAN MEHMETOGLU AND BOJAN POPOV If δ 0 < 0, then δ 0 + = 0 and there is nothing to prove. If δ 0 > 0, we proceed as in the proof of Step as follows,.4 δ 0 δ 0 =δ 0 + δ 0 δ 0 δ 0 =k0 n+ = k0 n+ v n+ v n+ v n+ +v0 n v n+ v n+, v n+ v n+ where k0 n+ := δ 0 + δ 0 > 0. We still need to check the sign of δ 0 δ 0 so we rewrite.0 for v n+ and subtract it from v 0 n to get v0 n v n+ vn 0 v + n γ + β + β + β + v0 n v 4 γ n.5 γ + β β + β + v0 n v n 4 γ. By similar arguments used in.9 and., we have.6 v n+ v n+ 8 + β c v 0, where c = if the assumption of Case holds, or c =+βγ /γ if the assumption of Case holds. Now we apply these bounds to δ 0 δ 0 : δ 0 δ 0 = v n+ v n+ +v0 n vn+ 8 + β c v 0 γ + β.7 + β + β + v0 n v n 4 γ 8 β γ + β c β + β + v0 n v n γ 0. Finally,.4 and.7 imply δ 0 + δ 0 0. This proves Step and Lemma.. By symmetric arguments it can also be shown that δ m + δ m 0and δ m + δ m 0. Gathering all the results we have established so far, we end up with.8 δ 0 + +δ + +δ m + +δ m + δ δ + + δ m + + δ m +, which leads us to m δ + =δ 0 + +δ + +δ m + +δ m + + δ +.9 =0 = m δ δ + + δ m + + δ m + + δ + = δ. =0 This completes the proof for m 4. Next, we consider the remaining cases. Case. m =. =

11 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES It is the same as Case except that there are no middle umps δ,...,δ m in.0. Hence, we have.0 δ 0 + +δ + +δ + +δ + δ 0 + δ + δ + δ. Case. m =. We have already proved that δ 0 + δ 0 0. By symmetric arguments it is also true that δ + δ 0. Next, we need to show that δ δ since both are positive. From.8 we have v n+ v n+ and by analogous arguments v n+ v n+. Together they imply δ = v n+ Therefore, we conclude that v n+ v n+ v n+ = δ.. δ 0 + +δ + +δ + δ 0 + δ + δ. Case 4. m =. We already have δ 0 + δ 0. We still need to show that δ + δ. If δ 0, then we are done. So suppose δ > 0 and observe that δ = v n+ and hence, v n+ v n+ v n+ = v n+ v n+. δ 0 + +δ + δ 0 + δ. Therefore, in all four cases m =,,, 4, we have. δ + δ. =0 =0 v n+ v n+ = δ, Now, we restrict the choice of σ in the modified minmod limiter see.0, to beliketheoneinmaprsee.. Namely,wedefinema, b in.0 with σ such that { a, a b,.4 sgσ =sgs, where s = b, b a. Under this assumption on the limiter, we have the following result. Theorem.4. Let u 0 L R, f be strictly convex in Ru 0 and f be bounded on R. Then there exists constant β which depends only on the ratio γ /γ and an absolute constant c, c 9000, such that under the CFL condition λ max f w β, w Ru 0 the NT scheme with the minmod limiter ma, b defined as in.0.4 satisfies the one-sided Lipschitz condition.4 provided that.5 min, c maxa +,b + σ. min a, b

12 ORHAN MEHMETOGLU AND BOJAN POPOV Proof. The case σ 0 follows from Theorem.. Hence, we only consider the case σ 0. This implies that we take a negative slope reconstruction at local minima and local maxima. Following the same steps as in the proof of Theorem., in the general case m 4 we get.6 δ δ 4 v 0, and δ 0 δ 0 0, where we recall that δ < 0 δ and σ<0 implies that δ δ.thus,.7 δ 0 + +δ + δ 0 + δ + 4 σ d, where d =min δ, δ.8 δ σ d + 6 σ d c. Notice that by.5, we obtain 5 6 δ + 4 δ δ c δ +δ δ, which implies.9 δ 0 + +δ + δ 0 + δ + c Δ δ +Δ δ. By symmetric arguments, it can also be shown that.40 δ m + +δ m + δ m + δ m + c Δ δm +Δ δm+. The remaining cases, m, can be handled in a similar way and we skip their proofs. We conclude that.4 δ + =0 =0 δ + c m Δ δ+ δ +, =0 where the last inequality follows from the one-sided stability result proven in [0] for any nonnegative ump sequence; see 56 on page 55 in [0]. Remark.5. If the local umps a, b in the minmod limiter.0.4 are such that c maxa +,b +.4 min a, b, we can recover the MAPR limiter taking σ as in.. Note that.4 is always true if c. However, even though the bound c 9000 can be improved, the current approach does not allow us to prove the one-sided Lipschitz condition.4 with c. Therefore, the minmod limiter.0.4 is more restrictive than MAPR in some cases. Analogous to [0], a maximum principle and a one-sided stability result implies a convergence result. To state the convergence theorem we briefly introduce the space of functions of bounded variation and one-sided Lipschitz classes which are used in the context of conservation laws. Definition.6. The space Lip, L R consists of all functions g L R such that the seminorm.4 g Lip,L R := lim sup gx + y gx dx y>0 y R is finite.

13 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES For functions g Lip, L R we consider the classes Lips, L p R+ defined by.44 g y g + Lp R My s, y > 0. The smallest M 0 for which.44 holds is denoted by g Lips,Lp R+. Whenwe set p = and s =, we obtain the class Lip,L R+, which is the usual onesided Lipschitz class used in conservation laws denoted by Lip+; see for example [8]. With this notation, by repeating exactly the same steps as in section 4 in [0], we obtain the following convergence result. Theorem.7. Let u 0 Lip, L R Lip, L R+. Then,thereexistsβ>0 such that under the CFL condition λ f L R β the NT scheme based on the limiter.0,.4.5 converges to the unique entropy solution of.. References [] Paul Arminon and Marie-Claude Viallon. Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d espace. C. R. Acad. Sci. Paris Sér. I Math., 0:85 88, 995. MR087 95:655 [] Ivan Christov and Boan Popov. New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. J. Comput. Phys., 7: , 008. MR d:650 [] Ami Harten, Börn Engquist, Stanley Osher, and Sukumar R. Chakravarthy. Uniformly highorder accurate essentially nonoscillatory schemes. III. J. Comput. Phys., 7: 0, 987. MR a:6599 [4] Ami Harten and Stanley Osher. Uniformly high-order accurate nonoscillatory schemes. I. SIAM J. Numer. Anal., 4:79 09, 987. MR a:6598 [5] G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher, and E. Tadmor. High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal., 56:47 68 electronic, 998. MR :6545 [6] Guang-Shan Jiang and Eitan Tadmor. Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput., 96:89 97 electronic, 998. MR f:658 [7] Alexander Kurganov and Eitan Tadmor. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys., 60:4 8, 000. MR d:655 [8] Haim Nessyahu and Eitan Tadmor. Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87:408 46, 990. MR i:6557 [9] Stanley Osher and Eitan Tadmor. On the convergence of difference approximations to scalar conservation laws. Math. Comp., 508:9 5, 988. MR m:65086 [0] Boan Popov and Ognian Trifonov. One-sided stability and convergence of the Nessyahu- Tadmor scheme. Numer. Math., 044:59 559, 006. MR c:65 Department of Mathematics, Texas A&M University, 68 TAMU, College Station, Texas address: omehmet@math.tamu.edu Department of Mathematics, Texas A&M University, 68 TAMU, College Station, Texas address: popov@math.tamu.edu