The moment of entropy equation for vector-BGK model results in the entropy equation for macroscopic model. However, this is usually not the case in numerical methods because the current literature consists only of entropy conserving/stable schemes for macroscopic model (to the best of our knowledge). In this paper, we attempt to fill this gap by developing an entropy conserving scheme for vector-kinetic model, and we show that the moment of this results in an entropy conserving scheme for macroscopic model. With the numerical viscosity of entropy conserving scheme as reference, the entropy stable scheme for vector-kinetic model is developed in the spirit of [33]. We show that the moment of this scheme results in an entropy stable scheme for macroscopic model. The schemes are validated on several benchmark test problems for scalar and shallow water equations, and conservation/stability of both kinetic and macroscopic entropies are presented.
We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. Entropy dissipation and additional coupling in the overlap region are added through the use o…
We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the…
Entropy-conserving numerical fluxes are a cornerstone of modern high-order entropy-dissipative discretizations of conservation laws. In addition to entropy conservation, other structural properties mimicking the continuous level such as pressure equilibrium and kinetic energy preservation are important. This note proves that there are no numerical fluxes conserving (one of) Harten's entropies for the compressible Euler equations that also preserve pressure equilibria and have a density flux independent of the pressure. This is in contrast to fluxes based on the physical entropy, where even kin…
This paper proposes a charge-conserving, variational, spatio-temporal discretization for the drift-kinetic Vlasov-Maxwell system, utilizing finite-elements for the electromagnetic fields and the particle-in-cell approach for the Vlasov distribution. The proposed scheme is fully electromagnetic, dealing with fields instead of potentials, and includes the effects of polarization and magnetization in the Gauss and Ampère-Maxwell laws, a consequence of reducing the full particle dynamics to drift-center dynamics. There is, however, no need to invert the Gauss law: it is satisfied automatically at …
Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum), pressure-robustness and $Re$-semi-robustness ($Re$: Reynolds number) are three important properties of Navier-Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum under some suitable senses; pressure-robustness means that the velocity errors are independent of the continuous pressure; $Re$-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on i…
GPU-Accelerated Energy-Conserving Methods for the Two-Dimensional Hyperbolized Serre-Green-Naghdi Equations
Massachusetts Institute of Technology（麻省理工学院）；Johannes Gutenberg University Mainz（美因茨约翰内斯古腾堡大学）；INRIA, U. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（法国国家信息与自动化研究所、波尔多大学、法国国家科学研究中心、波尔多国立理工学院、数学与交叉研究实验室）
In this paper, we develop a family of third order asymptotic-preserving (AP) and asymptotically accurate (AA) diagonally implicit Runge-Kutta (DIRK) time discretization methods for the stiff hyperbolic relaxation systems and kinetic Bhatnagar-Gross-Krook (BGK) model in the semi-Lagrangian (SL) setting. The methods are constructed based on an accuracy analysis of the SL scheme for stiff hyperbolic relaxation systems and kinetic BGK model in the limiting fluid regime when the Knudsen number approaches $0$. An extra order condition for the asymptotic third order accuracy in the limiting regime is…
Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow 0$). However, in the intermediate regime (say, $\varepsilon=O(Δt)$), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptoti…
We present an energy-conserving numerical scheme to solve the Vlasov-Maxwell (VM) system based on the regularized moment method proposed in [Z. Cai, Y. Fan, and R. Li. CPAM, 2014]. The globally hyperbolic moment system is deduced for the multi-dimensional VM system under the framework of the Hermite expansions, where the expansion center and the scaling factor are set as the macroscopic velocity and local temperature, respectively. Thus, the effect of the Lorentz force term could be reduced into several ODEs about the macroscopic velocity and the moment coefficients of higher order, which coul…
Projection-based model order reduction of an ordinary differential equation (ODE) results in a projected ODE. Based on this ODE, an existing reduced-order model (ROM) for finite volume discretizations satisfies the underlying conservation law over arbitrarily chosen subdomains. However, this ROM does not satisfy the projected ODE exactly but introduces an additional perturbation term. In this work, we propose a novel ROM with the same subdomain conservation properties which indeed satisfies the projected ODE exactly. We apply this ROM to the incompressible Navier-Stokes equations and show with…
We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transpor…
Structure-Informed Data-Driven Reduced-Order Modeling of Scalar Hyperbolic Conservation Laws via Kinetic Defect Measure
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin（德克萨斯大学奥斯汀分校航空航天工程与应用力学系）
The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g., for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step si…
A solution is proposed to a longstanding open problem in kinetic theory, namely, given any set of realizable velocity moments up to order 2n, a closure for the moment of order 2n+1 is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order (n $\le$ 2). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is define…
A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes for the full order model that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggere…
Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室与法国国家科学研究中心，皮埃尔和玛丽居里大学（巴黎第六大学））
The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a $L^2$ stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then a…
Stability Analysis of Quadrature-based Moment Methods for Kinetic Equations
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University（清华大学周培源应用数学研究中心）；Department of Energy and Power Engineering, Tsinghua University（清华大学能源与动力工程系）；Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）
Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems
Laboratoire de Mathématiques Jean Leray, Centre National de la Recherche Scientifique, Université de Nantes（让·勒雷数学实验室，法国国家科学研究中心，南特大学）；Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室，法国国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））
We propose and study a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L 2 relative entropy. Then, we adapt the arguments developed in [12] based the hypocoercivity method to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.
