A robust false transient method of lines for elliptic partial differential equations

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 (Schiesser, 1991), 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. Two examples are shown to demonstrate the robustness of the proposed method over the false transient method. ▸ A more robust method of solving elliptic PDEs is developed and discussed. ▸ A comparison of the false transient and the proposed method is explained. ▸ Several engineering/transport examples are considered. ▸ Linear solutions are described using matrix algebra and matrix exponentials. ▸ Nonlinear problems are solved, including unstable steady state solutions.