Finite Difference and Finite Volume Methods for Nonlinear Hyperbolic Systems and the Euler Equations

Abstract

Abstract In modern technologies one often encounters the necessity to solve compressible flow with a complicated structure. There are several conceivable models of compressible flow. Let us mention, for example, the model of inviscid, stationary, irrotational or rotational subsonic flow using the stream function formulation, and the models of transonic flow based on the small perturbation equation or full potential equation. There exists an extensive literature about the finite difference or finite element methods for the numerical solution of these models. (For a survey of mathematical and numerical methods for these models, see (Feistauer, 1998).) In a number of problems, the potential models are not sufficiently accurate, particularly in high speed (transonic or hypersonic) flow, because of the appearance of the so-called strong shocks with large entropy and vorticity production. This leads to the necessity of using the complete system of conservation laws consisting of the continuity equation, the Euler equations of motion and the energy equation (called the Euler equations in brief), which has been widely used during the last few decades for the modelling of flows in aeronautics, the aviation industry and steam or gas turbine design. Successively, the Euler equations have begun to be applied also to low Mach number problems on the one hand and to problems with chemical reactions on the other. These models neglect, of course, viscosity, but in many situations they give good results, reliable from the point of view of comparisons with experiments.