NUMERICAL SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF MIXED TYPE

Abstract

This chapter discusses a few recently developed numerical methods for the solution of nonlinear equations of mixed type. These methods have been used to calculate transonic flows with shock waves. These methods use finite difference approximations to the differential equation. Their formulation is based on an idea introduced by Murman and Cole, that is, to use central difference formulas in the subsonic zone, where the governing equation is elliptic, and upwind difference formulas in the supersonic zone, where it is hyperbolic. Thus, the numerical scheme has a directional bias. This corresponds to the upwind region of dependence of the flow in the supersonic zone and also serves the purpose of enforcing the entropy condition that discontinuous expansions must be excluded. The construction of a unified finite element method for the subsonic and supersonic zones appears difficult. The dominant term in the discretization error introduced by the upwind differencing acts like an artificial viscosity.