SIAM Journal on Numerical Analysis · 1998 · 33 citations · 33 references
Numerical AnalysisEngineeringVariational AnalysisScalar Conservation LawsNumerical FluxesComputational MechanicsPriori Error EstimatesNumerical ComputationNon-cartesian GridsNumerical SimulationRegularization (Mathematics)Approximation TheorySemi-implicit MethodHyperbolic Conservation LawInverse ProblemsGeneral TheoryNumerical Method For Partial Differential EquationNumerical TreatmentNumerical MethodsMultiscale Modeling
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 defined on non-Cartesian grids. We identify those schemes which are consistent and prove that the $L^\infty(0,T;L^1(\Bbb{R}^d))$-norm of the error goes to zero as $(\Delta x)^{1/2}$ when the discretization parameter $\Delta x$ goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of $(\Delta x)^{1/2}$ 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.
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