Numerical methods for heat equation with variable coefficients

Abstract

In this paper, two methods are developed for linear parabolic partial differential equation with variable coefficients, which are based on rational approximation to the matrix exponential functions. These methods are L-stable, third-order accurate in space and time. In the development of these methods, second-order spatial derivatives are approximated by third-order finite-difference approximations, which give a system of ordinary differential equations whose solution satisfies a recurrence relation that leads to the development of algorithms. These algorithms are tested on heat equation with variable coefficients, subject to homogeneous and/or time-dependent boundary conditions, and no oscillations are observed in the experiments. The method is also modified for a nonlinear problem. All these methods do not require complex arithmetic, and based on partial fraction technique, which is very useful for parallel processing.

Keywords:

• heat equation with variable coefficients
• finite differences
• method of lines
• non-linear heat equation
• rational approximation with distinct real poles

2000 AMS Subject Classification :

• 65M06
• 65N40

Acknowledgements

Authors are grateful to the anonymous referees for their careful reading of the paper and for valuable suggestions.