NUMERICAL SOLUTION OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF ELLIPTIC-HYPERBOLIC TYPE

Abstract

This chapter discusses the numerical solution of linear partial differential equations of elliptic-hyperbolic type. It reviews the numerical methods for the solution of linear equations of mixed type. In the theory of partial differential equations, there is a fundamental distinction between those of elliptic, hyperbolic, and parabolic type. Each type of equation has different requirements as to the boundary or initial data, which must be specified to assure existence, uniqueness, and continuous dependence on initial data, that is, for the problem to be well posed. These requirements are well known for an equation of any particular type. Also, many analytical and numerical techniques have been developed for solving the various types of partial differential equations, subject to suitable boundary conditions, including many nonlinear equations. The main body of results on the existence and uniqueness of solutions for the equations of mixed type have been primarily concerned with problems in two dimensions.