A Numerical Method of High Accuracy for Linear Parabolic Partial Differential Equations

Jun Liu,Yan Wang

doi:10.1155/2012/497365

Abstract

We report a new numerical algorithm for solving one-dimensional linear parabolic partial differential equations (PDEs). The algorithm employs optimal quadratic spline collocation (QSC) for the space discretization and two-stage Gauss method for the time discretization. The new algorithm results in errors of fourth order at the gridpoints of both the space partition and the time partition, and large time steps are allowed to save computational cost. The stability of the new algorithm is analyzed for a model problem. Numerical experiments are carried out to confirm the theoretical order of accuracy and demonstrate the effectiveness of the new algorithm.

Highlights

Quadratic spline collocation QSC is a kind of numerical methods for solving systems of differential equations, which gives rise to an approximate solution in the quadratic spline space
Based on the results {Cn}Nn 1 from scheme 2.24, we can obtain the approximate solution of system 2.1, 2.2, and 2.4 by {Un|Un Q0Cn}Nn 1
We can observe that the spectral radii of the matrix Q are smaller than those of Q for all values of σ. It seems that the QSC-TG algorithm is better than QSC-CN0 as far as stability is concerned

Summary

Introduction

Quadratic spline collocation QSC is a kind of numerical methods for solving systems of differential equations, which gives rise to an approximate solution in the quadratic spline space. An interesting property of the QSC method is that the optimal order of convergence can be obtained by adding appropriate perturbations to the spatial differential operators. Such an idea is used in the smooth cubic spline collocation for a special linear initial value problem 1. We can employ the optimal QSC method directly for the parabolic PDE and use high-order numerical methods for the resulting collocation equations, such as the two-stage Gauss method 15, 16. Based on QSC and the two-stage Gauss method, we can propose a QSC-TG algorithm in this paper for linear parabolic PDEs. The new algorithm gives high-order accuracy at the gridpoints of both the space partition and the time partition.

The QSC-TG Algorithm for Linear Parabolic PDEs

The Two-Stage Gauss Method for the Collocation Equation Denote the matrices

Stability of QSC-TG

Numerical Experiments

Conclusions