Structure-preserving discontinuous galerkin methods for multi-scale kinetic transport equations and nonlinear optics models

Abstract

To achieve this goal, one idea in the literatures involves an additional reformulation to the even-odd decomposition of the model: adding and subtracting a weighted diusive term in the decomposed system. We adopt this idea to the micro-macro decomposition of the model and develop a new family of high order AP schemes with unconditional stability in the diusive regime: IMEX-LDG scheme that involves implicit-explicit Runge-Kutta method in time and local DG method in space. AP property, uniform stability with respect to the Knudsen number and unconditional stability in the diusive regime are conrmed by asymptotic analysis and Fourier type stability analysis. Moreover, we discover a scaling structure, which guides the choice of the weight function in the additional reformulation.

In this thesis, we also design structure preserving schemes for a multiscale nonlinear optical model, the Kerr-Debye model with Lorentz dispersion, which describes the high frequency electromagnetic wave propagation through some optical media. It has three important physical structures. (1) As the optical media's relaxation time of the nonlinear response goes to zero, this model will converge to the Kerr model with Lorentz dispersion. (2) The third order susceptibility is nonnegative. (3) An energy relation is preserved. To preserve these physical structures, we design a few rst and second order in time AP nodal DG schemes. Our main contribution is to design a second order AP and positivity preserving modied exponential time integrator and an energy based approximation of the constitutive relation, which leads to provable energy preservation. The eect of the relaxation time of the nonlinear response is also numerically investigated with our new schemes.

Furthermore, we perform a rigorous energy type stability analysis for the first order in time IMEX1-LDG method. Our main contribution is a novel denition of the discrete energy and a careful exploration of the scattering operator. To the best of our knowledge, this is the rst rigorous proof of unconditional stability in the diusive regime together with uniform stability of schemes based on the additional reformulation strategy. The choice of the weight function in the additional reformulation is problem dependent and ad-hoc. A poor choice may lead to bad approximations. Hence, we further design a family of AP schemes called IMEX-DG-S scheme, which does not need any additional reformulation and avoids all weight related issues. At the same time, the IMEX-DG-S scheme still achieves uniform stability with respect to the Knudsen number and unconditional stability in the diusive regime. The key is that we design a new implicit-explicit temporal strategy and apply the Schur complement in the linear solver to improve the computational eciency. Asymptotic analysis, Fourier type stability analysis for high order schemes and energy stability analysis for the rst order IMEX1-DG1-S scheme are carried out to verify the AP and stability properties.

Many important physical phenomena have a multiscale nature, for example rareed gas dynamics, plasma physics, radiation transfer and nonlinear optics. For many of them, the multiscale nature is characterized by a characteristic parameter ", and as " ! 0, the underlying physical model will converge to a limiting model. To capture the multiscale behavior accurately and eciently, one attractive choice is the asymptotic preserving (AP) method, which preserves the asymptotic limit of the underlying physical system on the discrete level. Besides the asymptotic behavior, numerically preserving physical structures including energy relations and the positivity of some physical quantities is also desired. In this thesis, we develop and analyze structure preserving discontinuous Galerkin (DG) methods for multiscale problems, including a kinetic transport system under its diusive scaling and the Kerr-Debye model with Lorentz dispersion. We first consider a linear kinetic transport problem under a diusive scaling, which is a prototype model for radiation transfer and neutron transport. As the Knudsen number goes to zero, this model will asymptotically converge to a diusion model. In the diusive regime with small Knudsen number, many AP schemes have a parabolic time step restriction t = O(h2) (h is the mesh size). We want to enhance the stability and obtain unconditional stability in the diusive regime.