TY - JOUR

T1 - A priori error estimates for numerical methods for scalar conservation laws part III

T2 - Multidimensional flux-splitting monotone schemes on non-cartesian grids

AU - Cockburn, Bernardo

AU - Gremaud, Pierre Alain

AU - Yang, Jimmy Xiangrong

PY - 1998

Y1 - 1998

N2 - This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes denned on non-Cartesian grids. We identify those schemes which are consistent and prove that the L∞ (0, T; L1 (ℝd))-norm of the error goes to zero as (Δx)1/2 when the discretization parameter δx goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of (Δx)1/2 because (i) the conservation form of the schemes and (ii) the so-called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.

AB - This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes denned on non-Cartesian grids. We identify those schemes which are consistent and prove that the L∞ (0, T; L1 (ℝd))-norm of the error goes to zero as (Δx)1/2 when the discretization parameter δx goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of (Δx)1/2 because (i) the conservation form of the schemes and (ii) the so-called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.

KW - A priori error estimates

KW - Conservation laws

KW - Irregular grids

KW - Monotone schemes

KW - Supraconvergence

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U2 - 10.1137/S0036142997316165

DO - 10.1137/S0036142997316165

M3 - Article

AN - SCOPUS:0001180997

VL - 35

SP - 1775

EP - 1803

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 5

ER -