Second order accurate explicit finite volume schemes for the solution of Boltzmann-Peierls equation

Abstract

In this article we present the first and second order numerical schemes for the solution of initial value problems of the Boltzmann-Peierls equation (BPE). We also modify the numerical schemes for the solution of initial and boundary value problems (IBVP) of its derived hyperbolic moment system. BPE is an integro-differential equation which describes the evolution of heat in crystalline solids at very low temperatures. The BPE describes the evolution of the phase density of a phonon gas. The corresponding entropy density is given by the entropy density of a Bose-gas. We derive a reduced three-dimensional kinetic equation which has a much simpler structure than the original BPE, while it still retain all the properties of the original BPE. Using special coordinates, we get a further reduction of the kinetic equation in one space dimension. We introduce the discrete-velocity model of the reduce BPE in one space dimension. This discrete-velocity model can be discretized in space and time by using finite volume schemes. We derive both first and second order explicit upwind and central schemes for the discrete-velocity kinetic equation as well as for the derived moment system. We use the kinetic approach in order to prescribe boundary conditions for the IBVP of the moment system. Several numerical test cases are considered in order to validate the theory.