Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

Sparse-grid Discontinuous Galerkin Methods for the Vlasov-Poisson-Lenard-Bernstein Model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov-Poisson-Lenard-Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD

Nano–particle drag prediction at low Reynolds number using a direct Boltzmann–BGK solution approach

This paper outlines a novel approach for solution of the Boltzmann-BGK equation describing molecular gas dynamics applied to the challenging problem of drag prediction of a 2D circular nano–particle at transitional Knudsen number (0.0214) and low Reynolds number (0.25–2.0). The numerical scheme utilises a discontinuous-Galerkin finite element discretisation for the physical space representing the problem particle geometry and a high order discretisation for molecular velocity space describing the molecular distribution function. The paper shows that this method produces drag predictions that are aligned well with the range of drag predictions for this problem generated from the alternative numerical approaches of molecular dynamics codes and a modified continuum scheme. It also demonstrates the sensitivity of flow-field solutions and therefore drag predictions to the wall absorption parameter used to construct the solid wall boundary condition used in the solver algorithm. The results from this work has applications in fields ranging from diagnostics and therapeutics in medicine to the fields of semiconductors and xerographics

Core-Collapse Supernova Simulations With Spectral Two-Moment Neutrino Transport

The primary focus of this dissertation is to develop a next-generation, state-of-the-art neutrino kinetics capability that will be used in the context of the next-generation, state-of-the-art core-collapse supernova (CCSN) simulation frameworks \thornado\ and \FLASH.\index{CCSN} \thornado\ is a \textbf{t}oolkit for \textbf{h}igh-\textbf{or}der \textbf{n}eutrino-r\textbf{ad}iation hydr\textbf{o}dynamics, which is a collection of modules that can be incorporated into a simulation code/framework, such as \FLASH, together with a nuclear equation of state (EOS)\index{EOS} library, such as the \WeakLib\ EOS tables. The first part of this work extends the \WeakLib\ code to compute neutrino interaction rates from~\cite{Bruenn_1985} and produce corresponding opacity tables.\index{Bruenn 1985} The processes of emission, absorption, scattering of neutrinos from nucleons and nuclei, neutrino--electron scattering, and neutrino pair production and annihilation are included. The second part of this dissertation builds the special-relativity-corrected (\Ov) neutrino radiation module in \thornado, based on the spectral two-moment method.\index{\Ov} This part of the work involved studying the accuracy, efficiency, and robustness of the numerical solver. We propose a special kind of implicit-explicit scheme, PDARSs, based on efficiency, diffusion accuracy, and physics-preserving (positivity-preserving and realizability-preserving) requirements. \index{PD-ARS} Emission, absorption, scattering of neutrinos from nucleons and nuclei, neutrino--electron scattering, and neutrino pair production and annihilation are included as neutrino--matter couplings. The third part of this work builds interfaces between \FLASH\ and \thornado, \FLASH\ and \WeakLib, and \thornado\ and \WeakLib\ for simulations with the \FLASH\ hydrodynamics module, \WeakLib\ EOS module, and \thornado\ neutrino kinetics module. This part of the work includes data mapping between finite-volume grids and finite-element grids, time-step balancing between hydrodynamics time steps and radiation transport time steps, and GPU enhancement. The fourth part of this work makes a detailed comparison of the results of a spherically symmetric simulation performed by \FLASH+\thornado\ with the result of the \chimera\ code, which is a sophisticated, mature, and evolving code with spectral flux-limited diffusion (one-moment) neutrino kinetics and improved input physics~\citep{bruenn_etal_2020}. This part of the work demonstrates the ability of \FLASH+\thornado\ to perform CCSN simulations and quantifies the potential differences between the two codes caused by the different neutrino kinetics treatments, as well as other differences. Supported by all of the above work, spherically symmetric CCSN simulations with spectral two-moment neutrino kinetics were performed for three low-mass progenitors of 9-, 10-, and 11-Solar-mass (\solarmass) from~\cite{sukhbold_etal_2016}

An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics

In plasma simulations, where the speed of light divided by a characteristic length is at a much higher frequency than other relevant parameters in the underlying system, such as the plasma frequency, implicit methods begin to play an important role in generating efficient solutions in these multi-scale problems. Under conditions of scale separation, one can rescale Maxwell's equations in such a way as to give a magneto static limit known as the Darwin approximation of electromagnetics. In this work, we present a new approach to solve Maxwell's equations based on a Method of Lines Transpose (MOL$^T$) formulation, combined with a fast summation method with computational complexity $O(N\log{N})$, where $N$ is the number of grid points (particles). Under appropriate scaling, we show that the proposed schemes result in asymptotic preserving methods that can recover the Darwin limit of electrodynamics