We consider point sources in hyperbolic equations discretized by finite differences. If the source is stationary, appropriate source discretization has been shown to preserve the accuracy of the finite difference method. Moving point sources, however, pose two challenges that do not appear in the stationary case. First, the discrete source must not excite modes that propagate with the source velocity. Second, the discrete source spectrum amplitude must be independent of the source position. We derive a source discretization that meets these requirements and prove design-order convergence of the numerical solution for the one-dimensional advection equation. Numerical experiments indicate design-order convergence also for the acoustic wave equation in two dimensions. The source discretization covers on the order of $\sqrt{N}$ grid points on an $N$-point grid and is applicable for source trajectories that do not touch domain boundaries.
Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks ap…
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Center for Data and Simulation Science (CDS), University of Cologne（科隆大学数据与模拟科学中心）
The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $\mathcal{M}$ of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter $ h $, state-of-the-art discrete methods yield $ O(h) $ provable approximations. In this paper, we investigate the convergence of such approximations made by Manifold Moving Least-Squares (Manifold-MLS), a method that constructs an approximating manifold $\mathcal{M}^h$ using information from a given point cloud that was developed by Sober \& Levin …
Friedrichs' systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions
We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time-stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we…
The present article is devoted to developing the Legendre wavelet operational matrix method (LWOMM) to find the numerical solution of two-dimensional hyperbolic telegraph equations (HTE) with appropriate initial time boundary space conditions. The Legendre wavelets series with unknown coefficients have been used for approximating the solution in both of the spatial and temporal variables. The basic idea for discretizing two-dimensional HTE is based on differentiation and integration of operational matrices. By implementing LWOMM on HTE, HTE is transformed into algebraic generalized Sylvester e…
In this paper, a few dual least-squares finite element methods and their application to scalar linear hyperbolic problems are studied. The purpose is to obtain $L^2$-norm approximations on finite element spaces of the exact solutions to hyperbolic partial differential equations of interest. This is approached by approximating the generally infeasible quadratic minimization, that defines the $L^2$-orthogonal projection of the exact solution, by feasible least-squares principles using the ideas of the original $\mathcal{L}\mathcal{L}^*$ method proposed in the context of elliptic equations. All m…
Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at the same time observer-size macrostructures, multi-scale turbulent features, and also zero-scale shocks. Moreover, numerical methods for solving hyperbolic PDEs must reliably handle different families of waves: smooth rarefactions, and discontinuities of shock and contact type. In order to achieve these goals, an effective approach consists in the combination…
In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mecha…
Convexification Numerical Method for Imaging of Moving Targets
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Southern University of Science and Technology（南方科技大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Lanzhou University（兰州大学）
We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity--fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, \cred{a viscous fluid model}, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the norm…
Construction of reduced-order models (ROMs) for hyperbolic conservation laws is notoriously challenging mainly due to the translational property and nonlinearity of the governing equations. While the Lagrangian framework for ROM construction resolves the translational issue, it is valid only before a shock forms. Once that occurs, characteristic lines cross each other and projection from a high-fidelity model space onto a ROM space distorts a moving grid, resulting in numerical instabilities. We address this grid distortion issue by developing a physics-aware dynamic mode decomposition (DMD) m…
Adaptive mesh methods for hyperbolic conservation laws with bound-preserving flux limiters
South China University of Technology（华南理工大学）；University of Macau（澳门大学）；Guangzhou Nanfang College（广州南方学院）；Institute for Advanced Study, Beijing Normal-Hong Kong Baptist University（北京-香港浸会大学联合国际学院前沿研究院）
Linear models for the radiative transfer equation have been well developed, while nonlinear models are seldom investigated even for slab geometry due to some essential difficulties. We have proposed a moment model in MPN for slab geometry which combines the ideas of the classical PN and MN model. Though the model is far from perfect, it was demonstrated to be quite efficient in numerically approximating the solution of the radiative transfer equation, that we are motivated to further improve this model. Consequently we propose in this paper a new model following the chartmap in MPN with some s…
A system of high-order adaptive multiresolution wavelet collocation upwind schemes are developed for the solution of hyperbolic conservation laws. A couple of asymmetrical wavelet bases with interpolation property are built to realize the upwind property, and address the nonlinearity in the hyperbolic problems. An adaptive algorithm based on multiresolution analysis in wavelet theory is designed to capture moving shock waves and distinguish new localized steep regions. An integration average reconstruction method is proposed based on the Lebesgue differentiation theorem to suppress the Gibbs p…
Simple digital quantum algorithm for symmetric first order linear hyperbolic systems
Université du Québec, INRS-Énergie, Matériaux et Télécommunications（魁北克大学，INRS能源、材料与电信研究所）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Centre de Recherches Mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；School of Mathematics and Statistics, Carleton University（卡尔顿大学数学与统计学院）
