A novel partial differential algebraic equation (PDAE) solver: iterative space–time conservation element/solution element (CE/SE) method

Abstract

For solving partial differential algebraic equations (PDAEs), the space–time conservation element/solution element (CE/SE) method is addressed in this study. The method of lines (MOL) using an implicit time integrator is compared with the CE/SE method in terms of computational efficiency, solution accuracy and stability. The space–time CE/SE method is successfully implemented to solve PDAE systems through combining an iteration procedure for nonlinear algebraic equations. For illustration, chromatographic adsorption problems including convection, diffusion and reaction terms with a linear or nonlinear adsorption isotherm are solved by the two methods. The CE/SE method enforces both local and global flux conservation in space and time, and uses a simple stencil structure (two points at the previous time level and one point at the present time level). Thus, accurate and computationally-efficient numerical solutions are obtained. Stable solutions are guaranteed if the Courant–Friedrichs–Lewy (CFL) condition is satisfied. Solutions to two case studies demonstrate that the CE/SE numerical solutions are comparative in accuracy to those obtained from a MOL discretized by the 5th-order weighted essentially non-oscillatory (WENO) upwinding scheme with a significantly shorter calculation time.