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

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 4 th order scheme, when compared with the same produced, using the conventional 2 nd order scheme. The assessment is made, in terms of the discrete or norm of the true error of the converged numerical solution

Compact Schemes for the Spatial Discretization of Linear Elliptic Partial Differential Equations

A system of compact schemes used, to approximate the partial derivative 2 2 1 f x   and 2 2 2 f x   of Linear Elliptic Partial Differential Equations (LEPDE) ,on the non-boundary nodes, located along a particular horizontal grid line for 2 2 1 f x   and along a particular vertical grid line for 2 2 2 f x   of a two-dimensional structured Cartesian uniform grid. The aim of the numerical experiment is to demonstrate the higher order spatial accuracy and better rate of convergence of the solution, produced using the developed compact scheme. Further, these solutions are compared with the same, produced using the conventional 2 nd order scheme. The comparison is made, in terms of the discrete l l 2 &  norms, of the true error. The true error is defined as, the difference between the computed numerical and the available exact solution, of the chosen test problems. It is computed on every non-boundary node bounded in the computational domain.