Upwind scheme for non-hyperbolic systems

Abstract

Upwind scheme has been justified to be an accurate and stable numerical method to solve the hyperbolic system of equations. However, extension of an upwind scheme to solve a non-hyperbolic system may encounter the complex eigensystem in the coefficient matrix. It is difficult to determine the characteristic quantities by the complex eigenvectors and the upwind sense by the complex eigenvalues. Therefore, a suitable transformation is developed in the present work to derive a canonical form for the non-hyperbolic system in the real space. This canonical form will be identical with the characteristic equation if the system becomes hyperbolic. Based on this canonical equation, an upwind scheme can be constructed. This scheme is also extended to include the degenerate system and system with additional inter-drag and diffusion terms. Numerical treatment to avoid the impractically refined time step for a stable computation with a strong inter-drag term is also introduced. Normal-mode analyses are performed to indicate the stability of the proposed scheme and the associated time step constraint for a stable computation. Several representative model equations are solved and the calculated results show that the proposed scheme may be a useful tool to simulate both the hyperbolic and non-hyperbolic system of equations.