An efficient algorithm based on quadratic spline collocation and finite difference methods for parabolic partial differential equations

An efficient algorithm which combines quadratic spline collocation methods (QSC) for the space discretization and classical finite difference methods (FDMs), such as Crank-Nicolson, for the time discretization to solve general linear parabolic partial differential equations has been studied. By combining QSC and finite differences, a form of the approximate solution of the problem at each time step can be obtained; thus the value of the approximate solution and its derivatives can be easily evaluated at any point of the space domain for each time step.; There are two typical ways for solving this problem: (a) using QSC in its standard formulation, which has low accuracy O (h2) and low computational work. More precisely, it requires the solution of a tridiagonal linear system at each time step; (b) using optimal QSC, which has high accuracy O (h4) and requires the solution of either two tridiagonal linear systems or an almost pentadiagonal linear system at each time step. A new technique is introduced here which has the advantages of the above two techniques; more precisely, it has high accuracy O (h4) and almost the same low computational work as the standard QSC.