Numerical Solution of Symmetric Positive Differential Equations

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 with this method. 1. Symmetric Positive Linear Differential Equations. Let Q be a bounded open set in the m-dimensional space of real numbers, Rm. The boundary of Q will be Received May 18, 1967. Revised May 8, 1968. 763 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 764 THEODORE KATSANIS denoted by dQ, and its closure by Ö. It is assumed that du is piecewise smooth. A point in Rm is denoted by x = (x\, xi, • • •, xm) and an r-dimensional vector-valued function defined on Í2 is given by u = (u\, «2, • • -, ur). Also let a1, a2, ■ ■ -, am and G be given r X r matrix-valued functions and / = (/i, /2, • • •, fr) a given rdimensional vector-valued function, all defined on Ü (at least). It is assumed that the a* are piecewise differentiable. For convenience, let a = (a1, a2, • • -, am), so that we can use expressions such as (1.1) V.(«u)= Z¿-(«'«)• With this notation we can write the identity m « m t. i m ~ V f * \ _ 'S-* i T"1 * i=i ó\C¿ î=i OX i i=\ ox i simply as (1.2) V-(ow) = (V-a)ií + a-Vu . The definitions for symmetric positive operators and admissible or semiadmissible boundary conditions were introduced by Friedrichs [1]. Let K be the first-order linear partial differential operator defined by (1.3) Ku = a -Vu + V(aw) + Gu . K is symmetric positive if each component, a\ of a is symmetric and the symmetric part, (G + G*)/2, of G is positive definite on Ö. For the purpose of giving suitable boundary conditions, a matrix, ß, is defined (a.e.) on du by (1.4) ß = n-a, where n = (nh n2, • • -, nm) is defined to be the outer normal on d£2. The boundary condition Mu = 0 on dfi is semiadmissible if M = ß — ß, where ß is any matrix with nonnegative definite symmetric part, (ß + ß*)/2. If in addition, 3l(/¿ — |3) © 9I(m + ß) = Rr on the boundary, du, the boundary condition is termed admissible. (3l(^ — ß) is the null space of the matrix (ß — ß).) The problem is to find a function u which satisfies /X 5) Ku = f on Í2, Mm = 0 on du , where K is symmetric positive. Many of the usual partial differential equations may be expressed in this symmetric positive form, with the standard boundary conditions also expressed as an admissible boundary condition. This includes equations of both hyperbolic and elliptic type. However, the greatest interest lies in the fact that the definitions are completely independent of type. An example of potentially great practical importance is the Tricomi equation which arises from the equations for transonic fluid flow. The Tricomi equation is of mixed type, i.e., it is hyperbolic in part of the region, elliptic in part, and is parabolic along the line between the two parts. The significance of the semiadmissible boundary condition is that this insures License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SYMMETRIC POSITIVE DIFFERENTIAL EQUATIONS 765 the uniqueness of a classical solution to a symmetric positive equation. On the other hand, the stronger, admissible boundary condition is required for existence. The existence of a classical solution is generally difficult to prove for any particular case, and depends on properties at corners of the region. Let 3C be the Hilbert space of all square integrable r-dimensional vector-valued functions defined on 0. The inner product is given by (1.6) (u, v) = j u-v ,