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
SN - 0036-1429
VL - 35
SP - 1775
EP - 1803
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 5
ER -