TY - JOUR

T1 - A priori error estimates for numerical methods for scalar conservation laws. Part II

T2 - Flux-splitting monotone schemes on irregular Cartesian grids

AU - Cockburn, Bernardo

AU - Gremaud, Pierre Alain

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1997/4

Y1 - 1997/4

N2 - This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (Δcursive Greek chi)1/2 in L∞(L1) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p-consistent schemes, and prove that they converge to the entropy solution with the rate of (Δcursive Greek chi)min{1/2,p} in L∞(L1); again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (Δcursive Greek chi)1/2 in L∞(L1). We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

AB - This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (Δcursive Greek chi)1/2 in L∞(L1) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p-consistent schemes, and prove that they converge to the entropy solution with the rate of (Δcursive Greek chi)min{1/2,p} in L∞(L1); again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (Δcursive Greek chi)1/2 in L∞(L1). We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

KW - A priori error estimates

KW - Conservation laws

KW - Irregular grids

KW - Monotone schemes

KW - Supraconvergence

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U2 - 10.1090/s0025-5718-97-00838-7

DO - 10.1090/s0025-5718-97-00838-7

M3 - Article

AN - SCOPUS:0031527629

VL - 66

SP - 547

EP - 572

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 218

ER -