Von Neumann stable, implicit finite volume WENO schemes for hyperbolic conservation laws

Chieh-Sen Huang

Department of Applied Mathematics

National Sun Yat-sen University

huangcs@math.nsysu.edu.tw

    We present a new approach to defining implicit WENO (iWENO) schemes for systems of hyperbolic conservation laws. The approach leads to schemes that are simple to implement, high order accurate, maintain local mass conservation, apply to general computational meshes, and appear to be fairly robust. We present third and fifth order finite volume schemes in one and two space dimensions. We show that these iWENO schemes are unconditionally stable in the sense of von Neumann stability analysis, assuming the solution is smooth. The solution is approximated efficiently by two or three degrees of freedom per computational mesh element, independent of the spatial dimension. In space, the degrees of freedom are reconstructed implicitly to give high order approximation, while avoiding shocks and steep fronts due to the WENO framework. In time, high order quadrature is employed to produce a one step scheme. The approach is quite general, and we apply it to advection-diffusion-reaction equations with simple diffusion and reaction terms. Numerical results on nonuniform meshes in one and two space dimensions are presented. These explore the properties of the new schemes for solving hyperbolic conservation laws, advection-diffusion equations, advection-reaction equations, and the Euler system.