A Moving Mesh Numerical Method for Hyperbolic Conservation Laws

Abstract

Abstract. We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in i-'(R) to within 0(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 0(N~l). These numerical methods for conservation laws are the first to have proven convergence rates of greater than 0(N~l/2).

Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations

“…The rate of convergence has been tested for CFL numbers 0.625, 1.25, 2.5, 5 and 10 (only results for CFL numbers 0.625, 2.5 and 10 are shown in Tables 2 and 3). The results indicate that the method is first order accurate (even) for discontinuous solutions, which can be related to the fact that front tracking has a linear convergence rate [25]. We observe that for high CFL numbers the error on a coarse grid is dominated by the temporal splitting error.…”

“…We refer to [10,18,25] for a detailed description and analysis of the front tracking method. The fully discrete splitting approximation is now given by the formula…”

On Nonlinear Stochastic Balance Laws

“…Relevant continuous dependence results for deterministic conservation laws have been obtained in [17,1], and in [3] for strongly degenerate parabolic equations; see also [2,9]. We start with the following important lemma.…”

Numerical gradient methods for flux identification in a system of conservation laws

“…Notice by the way that, in the scalar case, it is in some sense Lipschitz continuous. Indeed, several authors, see for instance [10,11] for conservation laws, [12] for degenerate parabolic equations, provide L 1 dependence results of the entropy solution with respect to the flux. Application of Hölder's inequality then gives the result for J .…”

“…and this actually holds for scalar conservation laws; see [10,11]. On the other hand, if (c λ (L, t) − c(L, t))/λ has a limit, say δc, it turns out that δc has to solve the linearized equation…”