On Stability, Monotonicity, and Construction of Difference Schemes I: Theory
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
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A basic target algorithm for approximating unsteady hyperbolic conservation laws uses a finite volume formulation in three steps: recovery or reconstruction of a more accurate approximation from a set of cell averages; solution of the conservation law to obtain interface fluxes averaged over a time step; and computation of new cell averages at the new time level. In this paper the target is achieved by bringing together ideas from Brenier's transport collapse operator---using Lin, Morton, and Süli's Riemann--Stieltjes interpretation, van Leer's MUSCL algorithm, Colella and Woodward's PPM algorithm, and Goodman and LeVeque's flux approximation.First, second, and third order accurate algorithms are developed for nonuniform one-dimensional grids, and extensions are described for unstructured triangular meshes. The MUSCL-type scheme in one dimension is proved to be TV-stable right up to the natural CFL limit, in which characteristics cross no more than one cell in one time step, and under the least restrictive necessary TVD condition on the recovery stage.