Exact Flux Linearization for Convergence Improvement in the Implicit Godunov Method

Abstract

The present paper addressees the implicit Godunov method. A common way to handle the implicitness is to apply Newton’s iterations that require exact linearization of the residual. However, because the Godunov method adopts the exact, rather complicated, solution of the initial-value Riemann problem for the numerical flux, the linearization looks cumbersome. We show how to settle this problem and obtain the Godunov linearized flux in a compact analytical form. The linearization is based on the variational Riemann problem that is succeeded to solve analytically. The linear system of the Newton’s iteration is solved in two steps with the LU-SGS factorization method. Comparison with the method that uses conventional approximate flux linearization of Turkel and Jameson reveals certain advantages of the exact linearization in the rate of convergence.