Space-Time Finite Volume Differencing Framework for Effective Higher-Order Accurate Discretizations of Parabolic Equations

Abstract

A space-time finite volume differencing method is applied to construct new higher-order one-step and two-step implicit and explicit time integration schemes for parabolic equations. A unified space-time discretization error for approximating the integral form of the equation is first formulated by utilizing general weighted quadratures of neighboring grid points to approximate the flux integrals on space-time control volumes centered about each point. Efficient weighted quadrature approximations of the source term are then sought to account for local space-time flux approximations to all neighboring quadrature points on the control volume through a constrained minimization of the local error. Closed form descriptions of the weights that describe the resulting schemes and the leading coefficients of the residual errors are determined to guide the right selections of time step sizes to ensure consistent higher-order order convergence rates subject to stability requirements. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.