Robust False Transient Method of Lines
This page provides maple code that solves Elliptic PDE using the perturbation approach as described in the paper:
P. W. C. Northrop, P. A. Ramachandran, W. E. Schiesser, and V. R. Subramanian, "A Robust False Transient Method of Lines for Elliptic Partial Differential Equations," Chem. Eng. Sci., 90; 32–39 (2013).
Elliptic partial differential equations (PDEs) are frequently used to model a variety of engineering phenomena, such as steady-state heat conduction in a solid, or reaction-diffusion type problems. However, computing a solution can sometimes be difficult or inefficient using standard solvers. Techniques have been developed, including the method of lines, which can solve parabolic PDEs using well developed numerical solvers, but are not directly applicable to elliptic PDEs. The method of false transients overcomes this limitation by arbitrarily introducing a pseudo time derivative to modify the elliptic PDE to a parabolic PDE. However, this technique diverges for certain problems, such as when the solution is an unstable equilibrium point. A Jacobian-based perturbation approach is presented as an alternative for situations when the standard false-transient method fails.
The code found here solves Laplace's equation using both the standard false transient method and the perturbation approach as detailed in the above paper. The first figure details the convergence of the solution with respect to the pseudo time variable. The final figure shows the converged solution found using the perturbation approach. For Laplace's equation, there is little improvement for using the perturbation approach, however, for other equations the robustness of the perturbation approach allows problems to be solved that cause the false transient method to diverge.
In this code, the governing equations and boundary conditions can be directly changed. The number of node points can also be increased to provide increased accuracy, though this does increase the computational cost significantly. Also, the initial guess is provided, but can be changed. For certain problems, the initial guess can change the solution that is found, or even if a converged solution can be found, especially for the false transient method.
Please feel free to contact Dr. Venkat Subramanian for any comments.
List of other models: