A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach

Abstract

Abstract. In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the L 1 -contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.

Kruzkov’s estimates for scalar conservation laws revisited

“…It formalizes the method of B. Cockburn and P.-A. Gremaud [6] to replace the BV regularity of u by the BV regularity of v.…”

Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

“…This can be done with the help of a posteriori estimates. Such estimates are available, e.g., for finite difference schemes satisfying certain entropy inequalities [20] and for finite volume schemes [19,30], see also [11,12,26,39,48]. In particular consider first order schemes written in the (viscous) form…”

Finite volume relaxation schemes for multidimensional conservation laws

“…This is of some importance because the difficulties of applying Kruzkov's estimates to numerical schemes are highlighted. As noted first in [CG2], the classical approach of Kuznetsov is an "a posteriori" approach. This can be seen directly in the framework considered in this paper, simply by observing that the E−terms in the bound (4.5) depend only on the approximate solution u h .…”

“…An alternative "a priori" approach for deriving error estimates, which does not rely on the regularity properties of the schemes, was proposed and extensively analyzed in [CG2,3] for finite difference and in [CGY] for finite volume schemes. To carry out the program proposed in [CG1] one has to show an appropriate "discrete" stability for the scheme considered.…”

“…[CGY,Sections 2.a,b] for explicit assumptions. The development of ideas in [CG2,3], [CGY] and their application to relaxation schemes in unstructured meshes will be the subject of forthcoming work.…”