טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentPidgirnyak Anna
SubjectStability Analysis for a Family of Finite Difference
Approximations to Parabolic Systems
DepartmentDepartment of Applied Mathematics
Supervisor Professor Emeritus Moshe Goldberg


Abstract

The main purpose of this thesis is to discuss the L2-stability of a well-known family of finite difference schemes for  general, multi-space dimensional, well-posed parabolic initial-value problems of the form

                          

where u is the unknown vector, and Apq, Bp and C are constant matrices, Apq being Hermitian.

            The difference schemes are 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. For  we get the explicit Euler scheme, while all other values of θ yield implicit schemes. For  we obtain the Crank-Nicholson scheme, and  gives the Laasonen scheme.

            The first part of this work provides a survey of the background, the mathematical tools, and the literature pertaining to the subject matter. The heart of this survey consists of  a detailed discussion regarding the stability theorems of Goldberg (1998) and of Sun and Yuan (2000), which provide sufficient conditions for the stability of our difference schemes.

            The second part of this thesis discusses several new results, including proof of the optimality of the stability conditions of Goldberg and of Sun and Yuan. Another contribution, which seems worth mentioning, deals with the question of well- posedness of the approximated initial-value problem.

            The last chapter of this work provides several numerical calculations, which demonstrate the optimality of the stability conditions of Goldberg and of Sun and Yuan. These calculations fully support the theoretical results.