Local time stepping methods and discontinuous Galerkin methods applied to diffusion advection reaction equations
Kouevi, Assionvi Hove
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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.