Reduced proper orthogonal decomposition spectral element method for the solution of 2D multi-term time fractional mixed diffusion and diffusion-wave equations in linear and nonlinear modes

Abstract

At the present work, we propose a fast and high order numerical method for the solution of 2D multi-term time fractional mixed diffusion and diffusion-wave equation on regular and irregular convex and non-convex domains. We use a finite difference scheme based on a Crank-Nicolson method to discretize this equation in time variable and prove that the presented semi-discrete scheme is unconditionally stable. Due to the features of geometric flexibility and high order accuracy of spectral element method (SEM), this method is used to discretize the equation in space components. Afterwards, an error estimate of fully discrete scheme is given. In order to save the elapsed CPU time of simulation, a reduced-order model based on the proper orthogonal decomposition (POD) method is employed. Also, we provide the error estimate between the reduced POD-SEM solutions and usual SEM solutions. Moreover, we use the non-uniform temporal discretization to overcome the initial singularity. To investigate the accuracy of method, the numerical simulations are provided on regular and irregular convex and non-convex domains. Also, to illustrate the efficiency of proposed method, we compare the numerical solutions with the results of other methods developed in literature on regular and irregular domains from both accuracy and CPU time point of views.