Efficient Numerical Methods for Kinetic Equations with High Dimensions and Uncertainties

Yubo Wang, Purdue University

Abstract

In this thesis, we focus on two challenges arising in kinetic equations, high dimensions and uncertainties. To reduce the dimensions, we proposed efficient methods for linear Boltzmann and full Boltzmann equations based on dynamic low-rank frameworks. For linear Boltzmann equation, we proposed a method that is based on macro-micro decomposition of the equation; the low-rank approximation is only used for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second-order accurate (in both the fully kinetic and the limit regime) and asymptotic-preserving (AP). That is, in the diffusive regime, the scheme becomes a macroscopic solver for the limiting diffusion equation that automatically captures the low-rank structure of the solution. Moreover, the method can be implemented in a fully explicit way and is thus significantly more efficient compared to the previous state of the art. We demonstrate the accuracy and efficiency of the proposed low-rank method by a number of four-dimensional (two dimensions in physical space and two dimensions in velocity space) simulations. We further study the adaptivity of low-rank methods in full Boltzmann equation. We proposed a highly efficient adaptive lowrank method in Boltzmann equation for computations of steady state solutions. The main novelties of this approach are: On one hand, to the best of our knowledge, the dynamic lowrank integrator hasn’t been applied to full Boltzmann equation till date. The full collision operator is local in spatial variable while the convection part is local in velocity variable. This separated nature is well-suited for low-rank methods. Compared with full grid method (finite difference, finite volume,...), the dynamic low-rank method can avoid the full computations of collision operators in each spatial grid/elements. Resultingly, it can achieve much better efficiency especially for some low rank flows (e.g. normal shock wave). On the other hand, our adaptive low-rank method uses a novel dynamic thresholding strategy to adaptively control the computational rank to achieve better efficiency especially for steady state solutions. We demonstrate the accuracy and efficiency of the proposed adaptive low rank method by a number of 1D/2D Maxwell molecule benchmark tests. On the other hand, for kinetic equations with uncertainties, we focus on non-intrusive sampling methods where we are able to inherit good properties (AP, positivity preserving) from existing deterministic solvers. We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization problems. Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical multilevel Monte Carlo method especially for problems with discontinuities.

Degree

Ph.D.

Advisors

Lin, Purdue University.

Subject Area

Applied Mathematics

COinS