A High Order Finite Volume Method For One Dimensional Parabolic Integro-differential Equations

Finite volume method, also called as generalized difference method, is a numerical method for solving the differential equations. Due to its computation simplicity and preserving local conservation of certain physical quantities, it has been widely used in computing fluid mechanics,electromagnetic field and other fields.In the thesis, we consider a high order finite volume method for one dimensional parabolic integro-differential equations. The method is obtained by discretizing in space by arbitrary order vertex-centered finite volumes, followed by a modified Simpson quadrature scheme for the time stepping. Compared to the existed finite volume methods, this new finite volume scheme could achieve the desired accuracy with less data storage by employing higher-order trial spaces. The finite volume approximations are proved to possess optimal order convergence rates in the H1-and L2-norms, which are also confirmed by numerical tests.Firstly, we introduce the model of parabolic integro-differential equation and the idea of finite volume method, and expound the research status both at home and abroad. Secondly, we describe the finite volume methods for parabolic integro-differential equations. Then, we present some preliminary estimates of the Ritz-Volterra projection, the optimal order error estimates in the H~1- and L~2-norms are derived for both semidiscrete and fully discrete schemes. Finally, the theoretical results are verified by numerical examples.