A Class of High Resolution Schemes for Hyperbolic Conservation Laws and Convection--Diffusion Equations with Varying Time and Space Grids

Huazhong Tang and Gerald Warnecke

Abstract: This paper presents a class of 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 $L^\infty$ stable, satisfy 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 or 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 equations. 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.

Paper:
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Author(s):
Huazhong Tang, <hztang@math.pku.edu.cn>
Gerald Warnecke, <Gerald.Warnecke@Mathematik.Uni-Magdeburg.DE>
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