Higher order time integration of spatially nonhomogeneous Boltzmann equation: Deterministic and stochastic computations

Abstract

High accuracy of the time-differencing method for the spatially non-homogeneous Boltzmann equation, recently developed in [T. Ohwada, Journal of Compt. Phys., 139, 1 (1998)] as the second order approximation of the integral form of the equation along its characteristic line, is demonstrated numerically in both deterministic and stochastic computations of the standard Boltzmann equation for hard-sphere molecules. Comparisons are made with the results by the conventional splitting method and those by the splitting method with second order accurate collision step. Although the error of splitting method is reduced by the improvement of accuracy of the collision step, however, the accuracy is still first order because of the intrinsic error of the method. The intrinsic error is observed clearly in the stochastic computation of an initial value problem with a steep initial function, where the modified direct simulation Monte-Carlo proposed in the previous study is shown to work well.