We develop quadrature-based moment-closures for approximating kinetic Boltzmann equations. We first show how to replace the true distribution function by Dirac delta functions with variable weights and abscissas. We construct these functions to obtain a set of conditionally hyperbolic conservation laws. We then develop a high-order discontinuous Galerkin method to discretize the resulting systems with focus placed on limiters that guarantee that the numerical solutions remain in the convex hyperbolic regions of solution space. Numerical results in both 1D and 2D will be presented.

We propose several different approaches for solving fully implicit discretizations of a simplified Boltzmann-Poisson system with a linear relaxation-type collision kernel. This system models the evolution of free electrons in semiconductor devices under a low-density assumption. At each implicit time step, the discretized system is formulated as a fixed-point problem, which can then be solved with a variety of methods. A key algorithmic component in all the approaches considered here is a recently developed sweeping algorithm for Vlasov-Poisson systems. A synthetic acceleration scheme has been implemented to accelerate the convergence of iterative solvers by using the solution to a drift-diffusion equation as a preconditioner. The performance of four iterative solvers and their accelerated variants has been compared on problems modeling semiconductor devices with various electron mean-free-path.

Linear transport equations are ubiquitous in many application areas, including as a model for neutron transport in nuclear reactors and the propagation of electromagnetic radiation in astrophysics. The main computational challenge in solving the linear transport equations is that solutions live in a high-dimensional phase space that must be sufficiently resolved for accurate simulations. The three standard computational techniques for solving the linear transport equations are the (1) implicit Monte Carlo, (2) discrete ordinate \((\text{S}_N)\), and (3) spherical harmonic \((\text{P}_N)\) methods. We focus in this work on the \(\text{P}_N\) approximation, which is based on expanding the part of the solution that depends on velocity direction (i.e., two angular variables) into spherical harmonics. A big challenge with the \(\text{P}_N\) approach is that the spherical harmonics expansion does not prevent the formation of negative particle concentrations. In this work, we introduce an alternative formulation of the \(\text{P}_N\) approximation that hybridizes aspects of both \(\text{P}_N\) and \(\text{S}_N\). Although our basic scheme does not guarantee positivity of the solution, the new formulation allows for the introduction of local limiters that can be used to enforce positivity. We first develop our scheme and limiting strategy on the one-dimensional linear transport equations. We then show how to extend this idea to the multidimensional case using unstructured grids in phase space. The resulting scheme is validated on several standard test cases. Finally we develop a blended \(\text{P}_N\) and \(\text{H}^T_N\) scheme that hybridizes two approximations to improve the accuracy further.

The nonclassical linear Boltzmann equation was recently introduced to address transport problems taking place in certain types of random and heterogeneous systems. These include neutron transport in pebble bed and boiling water reactors, radiative transfer in the Earth’s cloudy atmosphere, and computer-generated imagery (CGI), to name a few. This nonclassical approach can properly capture and preserve important physical aspects of the system, enabling better mathematical and computational modeling of transport calculations in heterogeneous random media. In this talk, I will discuss a mathematical approach that allows one to numerically solve the nonclassical linear Boltzmann equation in a deterministic fashion using classical numerical procedures. The nonclassical transport equation describes particle transport for random statistically homogeneous systems in which the distribution function for free-paths between scattering centers is nonexponential. I will show how one can use a spectral method to represent the nonclassical flux as a series of Laguerre polynomials in the free-path variable s, resulting in a nonclassical equation that has the form of a classical transport equation. Numerical results validating the spectral approach will be given.

Low-rank approximation is a technique to approximate a tensor or a matrix with a reduced rank to reduce the memory required and computational cost for simulation. Its broad applications include dimension reduction, signal processing, compression, and regression. In this work, a dynamical low-rank approximation method is developed for the time-dependent radiation transport equation in slab geometry. Using a finite volume discretization in space and Legendre polynomials in angle we construct a system that evolves on a low-rank manifold via an operator splitting approach. We demonstrate that the low- rank solution gives better accuracy than solving the full rank equations given the same amount of memory.

The relativistic Vlasov-Maxwell System (RVM) models the behavior of collisionless plasma where the electrons interact via the electromagnetic fields they generate. In the RVM system, electrons could accelerate to significant fractions of the speed of light. Computationally, kinetic systems are difficult to solve due to their high dimensionality, and the RVM system introduces further difficulties such as a condition that operator splitting cannot be used to more easily handle the relativistic Lorentz force term. This renders conventional computing unable to efficiently resolve the RVM system. Thus we seek a high order alternative to operator splitting that performs well on manycore architectures. Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations (PDEs) with explicit time-stepping schemes, such as strong stability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL) argument requires. This increases the number of time steps needed to reach a target solution, which both reduces the efficiency of explicit time-stepping and severely damages the manycore scalability of the explicit method. In this work we discuss using the Regionally-Implicit discontinuous Galerkin (RIDG) method to solve the RVM system in two parts. First, we discuss the greatly enhanced manycore scalability of the RIDG method, which allows the method to outperform SSP-RKDG at scale. Secondly, we discuss a domain decomposition strategy for efficiently resolving the RVM system to high order accuracy.

We analyzed the superconvergence of DG method for scalar nonlinear conservation laws. We showed that the initial projection can be simplified in certain cases to achieve superconvergence. We also demonstrated that if piecewise polynomials of degree k are used, the superconvergence rate of the cell average of the DG solution is k+2 when sonic points appear in the computational domain, provided that certain spatial derivatives of the initial characteristic speed vanish at the sonic points.

Thin film equations result from the asymptotic expansion of the Navier-Stokes equation and are important in aircraft industry and several other applications. In this presentation we describe a DG approach to solving Thin Film equation. In particular we describe a nonlinear Local DG approach to solving a high order nonlinear operator. We use a Picard iteration approach to solving the nonlinear equations. We also present results on the unique behavior of the thin film equation.