High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation

Abstract

We apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a finite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb’s phenomenon with oscillations of size O(1) and recudes them to O(hr), where h is the mesh size and r is the order of accuracy. Our current implementation is three-dimensional, generalized for unequally spaced meshes, fully parallelized, and up to fifth order accurate (“WENO5”) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE’s in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, we need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, nonoscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes.