In this paper we develop a domain decomposition method (DDM), based on the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods, for solving multiscale problems involving macro sub-domains, where a macro model is valid, and micro sub-domains, where the macro model is not valid and a more costly micro model must be used. We take two examples, one from compressible gas dynamics where the micro sub-domains are around shocks, contacts and corners of rarefaction fans, and another one from semiconductor device simulations where the micro subdomains are around the jumps in the doping profile. The macro model is taken as the Euler equations for the gas dynamics problem and as a hydrodynamic model and a high field model for the semiconductor device problem. The micro model for both problems is taken as a kinetic equation. We pay special attention to the effective coupling between the macro subdomains and the micro sub-domains, in which we utilize the advantage of the discontinuous Galerkin method in its compactness of the computational stencil. Numerical results demonstrate the effectiveness of our DDM-DG method in solving such multi-scale problems.

Integration factor methods are a class of ''exactly linear part'' time discretization methods. In [Q. Nie, Y.-T. Zhang, R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006) 521-537], a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. In this paper, we solve this problem by applying the Krylov subspace approximations to the matrix exponential operator. Then we apply this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction-diffusion equations. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the method in resolving the stiffness of the DG spatial operator for reaction-diffusion PDEs. Application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown.

Integration factor methods are a class of “exactly linear part” time discretization methods. In [Q. Nie, Y.-T. Zhang, R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006) 521–537], a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent

Fast sweeping methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve the fast convergence for computations of static Hamilton-Jacobi equations. They take advantage of the properties of hyperbolic PDEs and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. The time-marching approach to steady state calculation is much slower than the fast sweeping methods due to the CFL condition constraint. But this kind of fixed-point iterations as time-marching methods have explicit form and do not involve inverse operation of nonlinear Hamiltonian. So it can solve general Hamilton-Jacobi equations using any monotone numerical Hamiltonian and high order approximations easily. In this paper, we adopt the Gauss-Seidel idea and alternating sweeping strategy to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations. Extensive numerical examples verify at least a 2 ∼ 5 times acceleration of convergence even on relatively coarse grids. The acceleration is even more when the grid is further refined. Moreover the Gauss-Seidel philosophy and alternating sweeping strategy improves the stability, i.e., a larger CFL number can be used. Also the computational cost is exactly the same as the time-marching scheme at each time step.