Ph.D Thesis

Ph.D StudentPidgirnyak Anna
SubjectStability of Finite Difference Approximations for Parabolic
Systems with Constant Coefficients
DepartmentDepartment of Applied Mathematics
Supervisor PROFESSOR EMERITUS Moshe Goldberg


This thesis deals with L2-stability of a well-known family of finite difference approximations to multi-space, well-posed parabolic initial-value problems of the form

where is an unknown m-vector, and Apq, Bp and C are constant matrices. Our family of difference schemes, known as the θ-method, is obtained by approximating the time derivative by a forward difference in time, whereas the spatial derivatives are approximated by a convex combination, with weights θ and , of centered differences taken at two adjacent time levels, and t. For  our approximation coincides with the explicit Euler scheme. All other values of θ yield implicit schemes; for  we get the Crank-Nicholson scheme, and  gives the Laasonen scheme.

The first part of this work provides a survey of the mathematical tools and the literature pertaining to the subject matter. The heart of this survey consists of a detailed discussion regarding the theory of Goldberg (1998) and of Sun and Yuan (2000), which offer sufficient stability conditions for our schemes in the classical case where the leading matrix coefficients Apq of the initial-value problem are Hermitian.

The second part of the thesis presents new results, which deal with the triangular case, in which the matrices Apq are upper triangular with real eigenvalues. We provide sufficient stability conditions for the θ-method, and show that the obtained conditions are optimal. We also exhibit numerical evidence which support the theory in both the Hermitian and the triangular cases.

We conclude by a brief discussion regarding the general case, where our initial-value problem cannot be reduced to either Hermitian or triangular form.