Compact Finite Volume Schemes For One Dimensional Linear And Semilinear Parabolic Equations With Third Boundary Conditions

The study of numerical solutions of partial differential equations plays an important rule in computational mathematics and finite difference, finite element and finite volume are three main methods. Compact finite difference refers to the kind of method with less stencil and high precision. Because of its simplicity and high accuracy, the method has been studied extensively. For one dimensional parabolic equation with third boundary conditions, this paper uses finite volume discretization method to construct a class of high accuracy finite volume scheme by absorbing the discretizing idea of compact difference method. We call it the compact finite volume scheme.Three chapters are included. Chapter one is an introduction of finite volume method and the research progress of compact difference method. This chapter also briefly describes the main structure of this paper. In chapter two, a compact finite volume scheme is presented for one dimensional parabolic equations with third boundary conditions. The linear algebraic system derived by this scheme has symmetrically tridiagonal property and can be solved by Thomas method. It is proved that the given scheme is convergent; with3.5-order accuracy in spatial direction and second-order accuracy in temporal direction with respect to discrete L2norm. Numerical examples illustrate that the theoretical analysis could be improved.In chapter three, based on the method of chapter two, a compact finite volume scheme is presented for one dimensional semilinear parabolic equations with third boundary conditions. In the process of constructing the scheme, we mainly solve two problems, one is that a new theoretical analysis is used to improve convergence order to4in spatial direction, and the other is making linearization for nonlinear term on the right side of equation, which finally constructed a compact finite volume scheme with fourth order accuracy in spatial direction and second-order accuracy in temporal direction. The linear algebraic system derived by this scheme has tridiagonal property and can be solved by Thomas method. Numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme.