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2003-12

A Class of High Resolution schemes for Hyperbolic conservation laws and Convection-Diffusion equations with varying time and space grids

by Tang, H., Warnecke, G..


Series: 2003-12, Preprints

MSC:
35L65 Conservation laws
65M06 Finite difference methods
65M99 None of the above, but in this section

Abstract:
This paper presents a class high resolution local time step schemes for
nonlinear hyperbolic conservation laws and the closely related convection-
diffusion equations, by projecting the solution increments of the underlying
partial differential equations (PDE) at each local time step. The main advantages
are that they are of good consistency, and it is convenient to implement
them. The schemes are L.. stable, statisfy a cell entropy inequality, and may
be extended to the initial boundary value problem of general unsteady PDEs
with higher-order spatial derivatives. The high resolution schemes are given
by combining the reconstruction technique with a second order TVD Runge-
Kutta scheme and a Lax-Wendroff type method, respectively.
The schemes are used to solve a linear convection-diffusion equation, the
nonlinear inviscid Burgers┬┤ equation, the one-and two-dimensional compressible
Euler equations, and the two-dimensional incompressible Navier-Stokes
equation. The numerical results show that the schemes are of higher-order
accuracy, and efficient in saving computational cost. The correct locations of
the discontinuities are also obtained, although the schemes are slightly
nonconservative.

Keywords:
Hyperbolic conservation laws, degenerate diffusion, high resolution scheme, finite volume method, local time discretization.