Hybrid discontinuous Galerkin method for the hyperbolic linear Boltzmann transport equation for multiscale problems.

Abstract

We propose an upwind hybrid discontinuous Galerkin (HDG) method for the first-order hyperbolic linear Boltzmann transport equation, featuring a flexible expansion suitable for multiscale scenarios. Within the HDG scheme, primal variables and numerical traces are introduced within and along faces of elements, respectively, interconnected through projection matrices. Given the variables in two stages, the HDG method offers significant flexibility in the selection of spatial orders. The global matrix system in this framework is exclusively constructed from numerical traces, thereby effectively reducing the degrees of freedom (DoFs). Additionally, the matrix system in each discrete direction features a blocked-lower-triangular stencil, enhancing the efficiency of solving hyperbolic equations through an upwind sweep sequence. Based on the proposed method, we perform an asymptotic analysis of the upwind-HDG method in the thick diffusion limit. The result reveals that the correct convergence of the upwind-HDG is closely associated with the properties of the response matrix L. A series of numerical experiments, including comparisons with the even-parity HDG, confirms the accuracy and stability of the upwind-HDG method in managing thick diffusive regimes and multiscale heterogeneous problems.