### 2003-12

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.