Model Reduction for Fisher's Equation with an Error Bound

Suchinthra Rungpitaxmana, Saifon Chaturantabut

Abstract

This work considers a model-order reduction (MOR) for Fisher's equation, which is generally used to describe many physical systems, such as chemical reactions, flame propagation, neurophysiology, nuclear reactors, and tissue engineering. Due to the nonlinearity in this type of system, solving the resulting discretized model for accurate solution could be time-consuming as the dimension gets large. Model-order reduction can be applied to improve the process of solving this large discretized model. In this work, a projection-based method called Proper Orthogonal Decomposition (POD) will be used first to project the state variables of the system on a low dimensional subspace, which will result in the decrease of unknowns in the systems. However, the computational complexity of the discretized nonlinear term still depends on the original large dimension. Discrete Empirical Interpolation Method (DEIM) is therefore used to eliminate this inefficiency. This POD-DEIM approach is applied on Fisher's equation with discontinuous initial conditions. An apriori error bound is derived for the approximations from POD-DEIM reduced system for the semi-implicit numerical scheme. The usefulness of this approach is illustrated through the parametric study of the varying boundary conditions. This work also investigates the effect of adding the snapshot difference quotients to construct basis sets used in POD and POD-DEIM reduced systems. The numerical results show that this POD-DEIM can substantially decrease the computational time while providing accurate numerical solution.