We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L¹(R) to within O(N- 2) by a piecewise linear function with $O(N)$ nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approximations, is accurate to O(N- 1). These numerical methods for conservation laws are the first to have proven convergence rates of greater than O(N- 1/2).
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Bradley J. Lucier (1986) studied this question.
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