Related papers: Solving Random Hyperbolic Conservation Laws Using …
Traditionally, classical numerical schemes have been employed to solve partial differential equations (PDEs) using computational methods. Recently, neural network-based methods have emerged. Despite these advancements, neural network-based…
This paper is concerned with the development of suitable numerical method for the approximation of discontinuous solutions of parameter-dependent linear hyperbolic conservation laws. The objective is to reconstruct such approximation, for…
Some hyperbolic systems are known to include implicit preservation of differential constraints: these are for example the time conservation of the curl or the divergence of a vector that appear as an implicit constraint. In this article, we…
We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the…
A method for the uncertainty quantification of nonlinear hyperbolic conservation laws with many uncertain parameters is presented. The method combines stochastic finite volume methods and tensor trains in a novel way: the dimensions of…
We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an…
We propose a Bayesian framework for feedback boundary control for hyperbolic balance laws. The method propagates a probability distribution over feedback parameters by using Lyapunov decay estimates as a likelihood. In the linear setting,…
We develop a method to learn physical systems from data that employs feedforward neural networks and whose predictions comply with the first and second principles of thermodynamics. The method employs a minimum amount of data by enforcing…
We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative…
Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a…
In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization…
We establish a general nonlocal approximation principle for the entropy solutions of scalar conservation laws on $\mathbb{R}$. More precisely, we show that the entropy solution to a nonnegative initial datum can be obtained as a weak-star…
A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries…
Conservation laws are an inherent feature in many systems modeling real world phenomena, in particular, those modeling biological and chemical systems. If the form of the underlying dynamical system is known, linear algebra and algebraic…
In this paper we study structure-preserving numerical methods for low Mach number barotropic Euler equations. Besides their asymptotic preserving properties that are crucial in order to obtain uniformly consistent and stable approximations…
We present a novel structure-preserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the Lax-Friedrichs flux to a high-order staggered…
For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for…
We study the following class of scalar hyperbolic conservation laws with discontinuous fluxes: \partial_t\rho+\partial_xF(x,\rho)=0. The main feature of such a conservation law is the discontinuity of the flux function in the space variable…
We study the convergence of a finite volume method based on the method of bicharacteristics for multidimensional hyperbolic conservation laws. In particular, we concentrate on the linear wave equation system and nonlinear Euler equations of…
The stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability of a non-linear scheme in terms of its cor- responding scheme in…