Difference Approximations for Inhomogeneous and Quasi-Linear Equations

Abstract

In particular, they showed that for a consistent approximation stability and convergence are equivalent. The results presented here have to do with the problem of approximating solutions of the inhomogeneous and quasi-linear equations which are obtained when terms of the form f(t) or g(t, u(t)) are added to equations such as the one above. Finite difference schemes for specific types of inhomogeneous equations and quasi-linear equations have been studied many times (see, for example, [1], [2], [4], and [7]). An extensive bibliography is included in [2]. In [7] Strang deals with linear inhomogeneous parabolic and hyperbolic partial differential equations; he shows that here, too, with appropriate definitions, stability and convergence are equivalent. Fritz John [4] deals with parabolic equations and includes a section on quasi-linear equations. John's paper is particularly remarkable in that he does not assume existence of solutions for the differential equations but proves existence using properties of the difference approximations. Frequently a difference approximation for an inhomogeneous equation or a quasi-linear equation is a simple modification of an approximation for the associated homogeneous equation. In this paper it will be shown that, it is often sufficient to consider the problem of convergence only for the homogeneous equation. That is, the conditions for convergence for linear homogeneous equations, which are provided by the Lax-Richtmnyer theory, are sufficient for the more general equations. Duhamel's method is used to show that a convergent approximation for a linear homogeneous equation