Numerical solution of symmetric positive differential equations

Abstract

A finite-difference method for the solution of symmetric positive linear differential equations is developed. The method is applicable to any region with piecevvise smooth boundaries. Methods for solution of the finite-difference equations are discussed. The finite-difference solutions are shown to converge at essentially the rate 0(h1'2) as h-> 0, h being the maximum distance between adjacent mesh-points. An alternate finite-difference method is given with the advantage that the finite-difference equations can be solved iteratively. However, there are strong limitations on the mesh arrangements which can be used with this method. Introduction. In the theory of partial differential equations there is a fundamental distinction between those of elliptic, hyperbolic and parabolic type. Generally each type of equation has different requirements as to the boundary or initial data which must be specified to assure existence and uniqueness of solutions, and to be well posed. These requirements are usually well known for an equation of any particular type. Further, many analytical and numerical techniques have been developed for solving the various types of partial differential equations, subject to the proper boundary conditions, including even many nonlinear cases. However, for equations of mixed type much less is known, and it is usually difficult to know even what the proper boundary conditions are. As a step toward overcoming this problem Friedrichs [1] has developed a theory of symmetric positive linear differential equations independent of type. Chu [2] has shown that this theory can be used to derive finite-difference solutions in two-dimensions for rectangular regions, or more generally, by means of a transformation , for regions with four corners joined by smooth curves. In this paper a more general finite-difference method for the solution of symmetric positive equations is presented (based on [3]). The only restriction on the shape of the region is that the boundary be piecewise smooth. It is proven that the finite-difference solution converges to the solution of the differential equation at essentially the rate 0(h112) as h-> 0, h being the maximum distance between adjacent mesh-points for a two-dimensional region. Also weak convergence to weak solutions is shown. An alternate finite-difference method is given for the two-dimensional case with the advantage that the finite-difference equation can be solved iteratively. However , there are strong limitations on the mesh arrangements which can be used

A Finite Difference Method for Symmetric Positive Differential Equations

“…This differs from usual finite difference procedures for boundary value problems, where the values on the boundary are known data. The finite difference method of Katsanis [7] is based on the formulation obtained by applying Green's theorem in any region in Q centered at each grid point. We shall see there exists a unique solution of (3.1), (3.2).…”

“…That is, We immediately have ||pauaIIo < I!uaIIa ' ^Vfi e ^a(^) • We shall need the following lemma which is given in [7].…”

“…Furthermore, it also gives a simple and unified numerical treatment for these problems (see e.g. [1,7,9]). Otherwise, if a numerical method applies directly to a PDE of mixed type, the treatment of the interface on which the PDE changes type is in general very difficult to handle.…”

“…Several numerical methods have been developed for Friedrichs's systems [7,9,10]. Friedrichs [5] was the first to propose a finite difference procedure for the numerical solutions of symmetric positive systems in rectangular regions.…”

“…Chu [4] further studied this method and extended it to curvilinear rectangular domains, but the rate of convergence was not established. Katsanis [7] gave a finite difference method for the Tricomi problem, using symmetric positive systems, which is applicable to any region with piecewise smooth boundaries, and showed that the rate of convergence is 0(hx/1), where h is the size of a regular mesh. However, he imposed on the system an extra constraint that the boundary matrix should be positive definite.…”