The heterogeneous multi-scale method based on the discontinuous Galerkin and finite volume schemes

This thesis contains three related topics. All topics arise from the numerical approximation for multi-scale problems.; In the first part, we develop a discontinuous Galerkin (DG) method, within the framework of the heterogeneous multi-scale method (HMM), for solving hyperbolic and parabolic multi-scale problems. Hyperbolic scalar equations and systems, and parabolic scalar problems are considered. Error estimates are given for the linear equations and numerical results are provided for the linear and nonlinear problems to demonstrate the capability of the method.; In the second part, the HMM method based on finite volume scheme is developed for solving hyperbolic problems. One dimensional hyperbolic scalar and systems are considered. Error estimates are given for the linear equations and numerical results are given for the linear and nonlinear hyperbolic problems.; The third topic is a domain decomposition method based on the discontinuous Galerkin method for kinetic/dynamic problems. An Euler system (macroscopic model) is used in the regions where the macroscopic model is valid and a BGK Boltzmann model (microscopic model) is used in the regions where the macroscopic model ceases to be valid, such as regions near a shock or a contact discontinuity. Stable and accurate interface coupling between the two models is explored. Numerical results are shown for a stationary shock, a moving contact discontinuity and a shock tube problem.