ERROR BOUNDS FOR THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BRADLEY J. LUCIER*

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1 EO BOUNDS FO THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BADLEY J. LUCIE* Abstract. Te expected error in L ) attimet for Glimm s sceme wen applied to a scalar conservation law is bounded by + 2 ) ) /2 T ) /2 u 0 BV ), were is te mes size and t is te time step. We sow tat te error in Godunov s sceme is bounded by te same expression and use tis result to analyze a variant of a large time step numerical metod of LeVeque tat may be viewed as a combination of Godunov s sceme and a metod of Dafermos. Glimm s Sceme. Glimm s sceme [3] is a probabilistic metod for proving existence of solutions for te yperbolic system of conservation laws, C) u t + fu) x =0, x,t>0, ux, 0) = u 0 x), x. Glimm sowed tat if te total variation of u 0 is sufficiently small, te equation is yperbolic and genuinely nonlinear in te sense of Lax, ten te approximate solution generated by is sceme converges almost surely to a weak solution of C); later Harten and Lax [5] sowed tat any Glimm weak solution satisfies te entropy condition. Glimm s sceme as also been applied wit some success as a numerical metod; see, for example, []. In tis note we bound te expectation of te L error of te approximate solution in te special case were C) is a scalar equation. In particular, we prove te following teorem. Teorem. If u n x, t) is te solution of Glimm s metod for t n t<t n+, ux, t) is te entropy solution of C), and T = N t, ten E u,t) u N,T) L )) + 2 ) ) /2 T ) /2 u 0 BV ), were is te mes spacing and t is te time step. A function is in BV ) if its derivative is a bounded measure. We remark tat Hoff and Smoller [7] ave derived certain error bounds for Glimm s metod using equidistributed rater tan random sequences of numbers. We note first tat altoug Glimm s sceme is usually defined on alternating meses te approximate solution is piecewise constant on te intervals [i, i +)) attimet n if n is even, and piecewise constant on [i /2), i +/2)) wenn is odd) tis in no way affects te error estimates given below. Consequently, a fixed mes is used for all time as a notational convenience. We prove our result for te following formulation of Glimm s sceme. We assume tat u 0 as bounded variation, and tat f L ) is finite. Coose a positive mes size. For eac integer i, let be [i, i+)), and let χ Ii be te caracteristic function of. Weassumetattetimestep, t, satisfies 0 < t * Department of Matematics, Purdue University, West Lafayette, Indiana. Tis material is based on work partially supported by te National Science Foundation under Grant No. DMS

2 /2 f L )), and we define t n = n t. For eac nonnegative integer n, we define a function U n : Z of bounded variation in te following way. Let ) Ui 0 = u 0 x) dx for eac integer i. IfUi n as been defined for all i, solve te initial value problem 2) u n t + fu n ) x =0, x,t n <t<t n+, u n x, t n )= U n i χ x), x. Coose a random variable X n+, uniformly distributed on [0,), so tat te set of random variables {X,...,X n+ } are independent; te values of U n+ i are ten given by 3) U n+ i = u n i + X n+,t n+ ) for every i. Corin [] seems to ave been te first to use exactly one random coice for all intervals. As can be seen from te definition, U n is itself a random variable tat depends on te sequence of random variables X,...,X n ; we propose to bound te expected value of te error at time t N, E u,t N ) u N,t N ) L )). Because, for any values of X troug X N, te approximate solution satisfies te differential equation exactly for x, t) t n,t n+ ), te following teorem of Kuznetsov applies see [8]). Lemma 2 [Kuznetsov]. If vx, t) is an exact solution of C) in every strip t n <t<t n+ tat is rigt continuous in t, andsup t [0,t N ] vt) BV ) is finite, ten 4) v,t N ) u,t N ) L ) u 0 v 0 L ) +2 u 0 BV ) N + ρ vt n ),ut n )) ρ vt n 0),ut n )) n= were ρ w, z) = x y η ) wx) zy) dx dy, andη is any nonnegative smoot function wit support in [, ] and integral one. From tis lemma it follows tat E u,t N ) u N,t N ) L )) u 0 ) u 0, 0) L ) +2 u 0 BV ) N + Eρ u n t n ),ut n )) ρ u n t n ),ut n ))). If we let E n f) denote te conditional expectation of f given X,...,X n,x n+,...,x N, ten writing t for t n, X for X n,andη x) for η x )), E n ρ u n t),ut)) ρ u n t),ut))) η x y) { u n i + X, t) uy, t) u n x, t) uy, t) } dx dy dx [0,) η x y) { u n z,t) uy, t) u n x, t) uy, t) } dz dx dy η x y) η z y)) 2 2 n= { u n z,t) uy, t) u n x, t) uy, t) } dz dx dy η x y) η z y) un z,t) u n x, t) dz dx dy. 2

3 If we now integrate over y, we find tat η x y) η z y) dy z x η L ). Trivially, u n z,t) u n x, t) u n t) BV Ii). So it follows tat 5) is bounded by 2 z x dz dx η L ) u n t) BV Ii) 2 η L ) u n t) BV Ii) 6 = 2 η L ) u n t) BV ) 6 Te inequality u n t) BV ) u 0 BV ) is clear, because te coice of te initial data ), te evolution of u n troug C) 2), and te random coice process 3) are all variation diminising. Tus, E n ρ u n t),ut)) ρ u n t),ut))) 2 6 η L ) u 0 BV ) uniformly wit respect to te oter random variables X i, implying tat Eρ u n t),ut)) ρ u n t),ut))) is bounded by te same quantity. Terefore, if T = N t, by using an obvious bound for te initial error, we ave 6) E u,t) u N,T) L )) u 0 BV ) +2 u 0 BV ) + T 2 t 6 η L ) u 0 BV ) By letting η 2 χ [,], η L ) may be cosen arbitrarily close to. Minimizing 6) wit respect to gives 7) E u,t) u N,T) L )) + 2 ) ) /2 T ) /2 u 0 BV ). Te teorem is proved. We remark tat if one cooses to interpret Glimm s metod as providing tat te approximate solution is equal to Ui n on [i, i +)) [t n,t n+ ], ten te above inequality still olds wit a small cange for te error incurred in at most one time step. It is well known tat monotone finite difference scemes, suc as Godunov s, perform better for problems wit uniformly convex fluxes tan for problems wit linear fluxes; in fact, Godunov s metod is O) accurate for te problem 8) u t + u2 2 ) x =0, x,t>0, ux, 0) = χ,0] x). For problems, suc as tis one, wose solution consists of a single sock of eigt one, te expected error in Glimm s sceme may be estimated directly by applying te central limit teorem. If te sock speed is s, and p = s t/, tenaftern time steps, te probability distribution of te sock location error is approximately normal wit mean 0 and variance σ 2 = Np p) 2. Asymptotically, te expected value of te L ) error, wic is te absolute value of te sock location error, is 8 π σ,or8 π p p) t )/2 T) /2,wereT = N t. Our bound on te ratio t/ implies tat 0 p /2. For example, wen Glimm s sceme is applied to 8) wit t = /2, p =/4, te expected value of te error is about t )/2 T ) /2. Teorem gives a bound of.547 t )/2 T ) /2 independently of te value of p, a fairly close result. 3

4 Godunov s Metod. Godunov s finite difference sceme [4] falls into te class of monotone, conservative, finite difference scemes for C), a class tat as been analyzed previously; see, for example, Kuznetsov s paper or Sanders [3]. Tese papers sow tat te error at time T is bounded by C +T ) /2 ) u 0 BV ) wentetimestep t is proportional to, but a good estimate of te constant is not available. In te next teorem we derive a different bound in wic te independent effects of t and appear along wit an explicit determination of a constant C, a result tat will prove useful in te next section. Godunov s sceme differs from Glimm s sceme only in tat Godunov determines U n+ i by averaging u n,t n+ )over : 3 ) U n+ i = 0 u n i + X, t n+ )dx. Te following teorem sows tat te error in Godunov s metod is bounded by te same expression tat bounds te expected error in Glimm s sceme. Teorem 3. If u n x, t) is te solution of Godunov s metod for t n t t n+, ux, t) is te entropy solution of C), and T = N t, ten 9) u,t) u N,T) L ) + 2 ) ) /2 T ) /2 u 0 BV ). Proof. We proceed as in Teorem. Again, wit t = t n, ρ u n t),ut)) ρ u n t),ut)) η x y) { Ui n uy, t) u n x, t) uy, t) } dx dy { } η x y) u n i + X, t) dx uy, t) [0,) un x, t) uy, t) dx dy η x y) { u n i + X, t) uy, t) u n x, t) uy, t) } dx dy dx. [0,) One may now follow te series of inequalities in 5) and te subsequent arguments to obtain te estimate in te statement of te teorem. Te estimate is rater sarp: by exploiting te relationsip of te solution of Godunov s metod and te cumulative probability distribution of a binomial random variable, one can sow tat Godunov s sceme applied to te problem wit fu) =u, u 0 x) =χ,0] x), and t/ =/2 as an asymptotic error rate of 4 π )/2 T) /2 =.283T) /2, compared to our estimate of T)/2 =.6330T) /2. Tat te error in tis example is O /2 ) may be inferred from results in [6].) Note tat our analysis applies even if te CFL condition t </2 f L )) is violated, as long as te wave interactions in te solution of 2) are calculated exactly; tis will prove important in te next section. LeVeque s Metod. LeVeque [] proposed a numerical sceme for scalar conservation laws in one space dimension wose main idea is to approximate te solution of C) using piecewise constant states, and to calculate te trajectories of socks straigt lines) and teir interactions exactly. Expansion waves are approximated by a series of constant states separated by entropy violating socks. At te end of eac time step te approximate solution is projected back onto te grid by averaging, as in Godunov s metod. LeVeque 4

5 as sown tat a variation of is metod is total variation diminising and tat a subsequence of numerical approximations converges to a weak solution of te conservation law [9]. He as also conjectured tat if entropy satisfying solutions of C) advance te approximate solution from one time step to te next, ten te weak solution to wic te approximations converge would indeed be te entropy weak solution. Among oter tings, we sow below tat even if certain entropy violating weak solutions are used to propagate te solution between time steps, te numerical approximations still converge to te entropy solution of C), and we obtain realistic estimates for te error. LeVeque s later extensions of is metod to systems of equations cose to ignore te wave interactions rater tan to calculate tem exactly [0]. Entropy violating socks used to model expansion waves may be cosen in suc a way tat LeVeque s formulation is equivalent to a metod of Dafermos [2], wic uses a piecewise linear approximation f to te flux f to advance te approximate solution between time steps. Altoug true sock speeds in te two metods differ speeds differ by less tan O) for weak socks and O 2 ) for strong socks), te numerical results of te two metods are qualitatively indistinguisable. Te sceme tat we analyze is as follows. Assume tat te flux f is C 2 and tat te initial data u 0 is in BV ) and is constant outside some finite interval. Let >0 be te mes size and t >0betetime step; define t n = n t. Let f be te continuous, piecewise linear interpolant of f wit breakpoints at i, for i Z. It is easily seen tat f f Lip 2 f L ), were g Lip =sup gx) gy) x y. x y Define u 0 by 0) u x, 0) = u 0 x) = u 0 σ) dσ χ Ii x). Te complete approximation u x, t) is as follows. For every n solve ) u t + f u ) x =0, x,t [t n,t n+ ), wit 2) u x, t n+ )= u σ, t n+ 0) dσ χ Ii x). Te analysis of te sceme is in two parts, corresponding to te two equations ) and 2). First we describe more carefully ow to solve ). In [2] Dafermos gives te entropy solution of ) as follows. He first reduces te problem to a iemann problem because te initial datum is piecewise constant. If u 0 is specified as { ul, if x 0, u 0x) = u r, if x>0. wit u l <u r, ten te vertices of te boundary of te convex ull of {u, v) u l u u r,v f u)} will consist of a set of points u l,f u l )), u,f u )),...,u k,f u k )), u r,f u r )), were {u i } is a linearly 5

6 ordered subset of {j}. Te solution will ten be given by te following set of constant states: u x, t) = u l, for < x t f u ) f u l ) u u l, u, for f u ) f u l ) u u l < x t f u 2 ) f u ) u 2 u,. u k, for f u k ) f u k ) u k u k < x t f u r ) f u k ) u r u k, u r, for f u r ) f u k ) u r u k < x t <. A similar result may be inferred if u l >u r by considering te convex ull of te set {u, v) u r u u l,v f u)}. Tus, te general solution of ) is found as a composition of iemann problems, all of wose solutions are socks. Te calculation is started anew wenever two socks coalesce into one. We now use te following teorem. Teorem 4 [2]). If f and g are Lipscitz continuous, u 0 and v 0 are in BV ), andu and v are te entropy solutions of te equations u t + fu) x =0, x,t>0, ux, 0) = u 0 x), x, and respectively, ten v t + gv) x =0, x,t>0, vx, 0) = v 0 x), x, ut) vt) L ) u 0 v 0 L ) + t f g Lip min u 0 BV ), v 0 BV ) ). Tus, if ū is te solution of ū t + f ū) x =0, x,t>0, ūx, 0) = u 0x), x, ten ) t ūt) ut) L ) 2 f L ) + u 0 BV ). Because u is te Godunov approximation to ū and u 0 BV ) u 0 BV ), it follows from te triangle inequality and 9) tat ut ) u T ) L ) + T 2 f L ) + 2 ) ) /2 T ) /2 u 0 BV ). Tis result implies tat if t is about te same size as, ten te error at time T is O+T +T ) /2 ); for t in tis regime, te averaging error of Godunov s metod dominates te error bound, causing te sceme to be O /2 ) accurate. However, if only one time step is taken, and t = T, ten te error at time T is O + T ), i.e., te metod is first order accurate. Tis explains wy LeVeque acieved suc good numerical results wen e reduced te number of time steps in is experiments []. 6

7 EFEENCES [] A. J. Corin, andom coice solution of yperbolic systems, J. Comp. Pys., ), [2] C. M. Dafermos, Polygonal approximations of solutions of te initial value problem for a conservation law, J. Mat. Anal. Appl., ), [3] J. Glimm, Solutions in te large for nonlinear yperbolic systems of equations, Comm. Pure App. Mat., 8 965), [4] S. K. Godunov, A finite difference metod for te numerical computation of discontinuous solutions of te equations of fluid dynamics, Mat. Sb., ), [5] A. Harten and P. D. Lax, A random coice finite difference sceme for yperbolic conservation laws, SIAMJ. Numer. Anal., 8 98), [6] G. Hedstrom, Models of difference scemes for u t + u x =0by partial differential equations, Mat. Comp., ), [7] D. Hoff and J. Smoller, Error bounds for te Glimm sceme for a scalar conservation law to appear). [6] N. N. Kuznetsov, Accuracy of some approximate metods for computing te weak solutions of a first-order quasi-linear equation, USS Comp. Mat. and Mat. Pys., 6 976), [9]. J. LeVeque, Convergence of a large time step generalization of Godunov s metod for conservation laws, Comm. Pure Appl. Mat., ), [0], A large time step generalization of Godunov s metod for systems of conservation laws to appear). [], Large time step sock-capturing tecniques for scalar conservation laws, SIAM J. Numer. Anal., 9 982), [2] B. J. Lucier, A moving mes numerical metod for yperbolic conservation laws, Mat. Comp. to appear). [3]. Sanders, On convergence of monotone finite difference scemes wit variable spatial differencing, Mat. Comp., ),