A MODIFIED LEAST-SQUARES FINITE ELEMENT METHOD FOR SOLVING THE BOLTZMANN TRANSPORT EQUATION

摘要

In this paper, we present a modified least-squares finite element method for solving theBoltzmann transport equation with spherical harmonics angular approximation (LSPN).Traditional LSPN methods usually adopt weak form boundary condition which containsuncertain parameter and it may lead to inaccurate results without proper scaling for problemswith sharp removal cross section interfaces. In this work, we derived the Euler-Lagrangeequation of the least-squares weak form and found that in order to eliminate the discretizationerror at material interface, the weak form of the first-order transport equation need to besubtracted. In this way, the natural boundary condition can be applied directly. In order toimprove the convergence rate and keep the stability of the equation, a scaling parameter needto be multiplied to the least-squares term, currently we choose the scaling parameter as thecharacteristic length of element which is determined by experience. In fact, the scalingparameter has very significant impact on the solver convergence rate and the stability, theoptimal value for the scaling parameter need further studies. We test our method with somebenchmark problems and the numerical results show it has good accuracy, stability and voidapplicability.