Two Kinds Of High Accuracy Finite Volume Methods For Two-Point Boundary Value Problems

This paper studies two kinds of high accuracy finite volume methods for two-point boundary value problems and their convergence analysis which includes two chapters. In chapter one, we present a superconvergent finite volume element method for mixed two point boundary value problems based on mixed cubic interpolations. The linear algebraic system derived by this scheme has pentadiagonal property and can be solved by band eliminition method. It is proved that the given scheme is convergent with fourth-order accuracy according to H1discrete seminorm. Finally, a numerical example is given to show the correctness of theoretical analysis of the scheme by implementing Fortran codes. We also show the numerical derivatives have fourth order accuracy at the superconvergent points of the interpolation.In chapter two, a compact finite volume scheme is presented for two point periodic boundary value problem. The linear algebraic system derived by this scheme has periodic tridiagonal property. By consructing a transformation, we obtain two linear algebraic systems which have tridiagonal property and can be solved by Thomas method, so we can improve calculation efficiency. It is proved that the given scheme is convergent with fourth order accuracy with respect to discrete H1semi-norm by energy method and can reach up to sixth order accuracy by using Richardson extrapolation technique. Furthermore, the post-processing formulas for the numerical value and derivative at the midpoint of every element are obtained, which have fourth order accuracy. Numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme and its extrapolation.