Asymptotic preserving space-time discontinuous Galerkin methods for a class of relaxation systems
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Various models derived from the Boltzmann equation can be used to model heat conduction, neutron transport, and gas dynamics. These models arise when one expands the distribution function for the Boltzmann equation in spherical harmonics, which results in singularly perturbed hyperbolic systems scaled by a diffusive relaxation parameter epsilon. In the diffusive limit, the perturbed equations limit to parabolic-type systems such as the heat equation and the advection-diffusion equation. Much work has been done in developing numerical schemes that are useful in the rarefied regime and that preserve the diffusive limit at the discrete level. A method that does this successfully is called asymptotic preserving (AP).
One difficulty in using standard numerical methods for these models is that they have a very restrictive CFL condition on time-step size which vanishes in the diffusive limit. Some attempts to overcome this hurdle have used implicit time-integration schemes, but this requires computing the solution to very large linear systems at each time step.
Our strategy is to develop a space-time discontinuous Galerkin method that will admit a less restrictive limit on the time-step size and will maintain its order of accuracy as the diffusive relaxation parameter becomes very small.