A Fast and Stable Algorithm for Linear Parabolic Partial Differential Equations

Abstract

We report a new algorithm for solving linear parabolic partial differential equations (PDEs) in one space dimension. The algorithm employs optimal quadratic spline collocation (QSC) for the space discretization and deferred correction (DC) for the time discretization. We can obtain a numerical solution with theoretical convergence order \(\mathcal {O}({\Delta } x^{4}+{\Delta } t^{\min [k_{0}+1,N_{m}]})\) by solving k0+1 tridiagonal linear systems at each time step, where k0 is the number of deferred corrections per time step and Nm is usually an integer which is not bigger than 8. The stability properties of the new algorithm are analyzed, and compared with the QSC-CN0 algorithm [5], which is a very efficient algorithm. Numerical experiments are attached to demonstrate the effectiveness of the new algorithm.