Local time stepping methods and discontinuous Galerkin methods applied to diffusion advection reaction equations

Abstract

Partial differential equations (PDEs), especially the diffusion advection and re action equations (DAREs), are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Due to the difficulty of finding exact solutions, developing efficient numerical methods for simulating the solution of the DAREs is a very important and challenging research topic. In this work, we present the transformation of the DAREs to ordinary differen tial equations (ODEs) using the standard finite element (FE) or the discontinuous Galerkin (DG) spatial discretization method. The resulting system of ODEs is then solved with standard time integrators such as implicit Euler methods, integrat ing factor method, exponential time differencing methods, exponential Rosenbrock methods, orthogonal Runge-Kutta Chebyshev methods. To illustrate the limitations of the FE method, we simulate and invert the cyclic voltammetry models using both spatial discretization methods (i.e. FE and DG) and show numerically that DG is more efficient. In many physical applications, there are special features (such as fractures, walls, corners, obstacles or point loads) which globally, as well as locally, have important effects on the solution. In order to efficiently capture these, we propose two new numerical methods in which the mesh is locally refined in time and space. These new numerical methods are based on the combination of the DG method with local time stepping (LTS) approaches. We then apply these new numerical methods to investigate two physical problems (the cyclic voltammetry model and the transport of solute through porous media). These numerical investigations show that the combination of the DG with the LTS approaches are more efficient compared to the combination of DG with standard time integrators.