Higher Order Compact Schemes for the Spatial Discretization of Linear Parabolic Partial Differential Equations

Abstract

A system of compact schemes used, to approximate the partial derivative of Linear Parabolic Partial Differential Equations (LPPDE), on the non-boundary nodes. Euler time integration for the temporal derivative, the Crank-Nicholson (CN)scheme for the spatial derivatives and the source term are used. The higher order spatial accuracy of the developed, central difference based compact schemes for the one and two-dimensional diffusion equations are validated, by solving numerically two test problems. The compact scheme based calculations involve a tridiagonal matrix vector multiplication and a vector vector subtraction. The interest is to demonstrate the higher order spatial accuracy and the better rate of convergence of the solution produced using the developed compact 4th order scheme, when compared with the same produced, using the conventional 2nd order scheme. The assessment is made, in terms of the discrete or norm of the true error of the converged numerical solution.