Related papers: On numerical methods for hyperbolic PDE with curl …

In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. (JCP 2000, JCP 2002), which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type…

Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems.…

We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems and which was…

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…

Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.…

In this paper, we present a semi-implicit numerical solver for a first order hyperbolic formulation of two-phase flow with surface tension and viscosity. The numerical method addresses several complexities presented by the PDE system in…

In this work, we introduce two novel reformulations of a recent weakly hyperbolic model for two-phase flow with surface tension. In the model, the tracking of phase boundaries is achieved by using a vector interface field, rather than a…

In this paper, we present a numerical homogenization scheme for indefinite, time-harmonic Maxwell's equations involving potentially rough (rapidly oscillating) coefficients. The method involves an $\mathbf{H}(\mathrm{curl})$-stable,…

For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy, including the Willmore and the Helfrich flows, we consider a numerical approach. In this study, we construct a structure-preserving…

This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87…

We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution,…

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the…

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…

Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to…

Machine learning techniques are being used as an alternative to traditional numerical discretization methods for solving hyperbolic partial differential equations (PDEs) relevant to fluid flow. Whilst numerical methods are higher fidelity,…

We study the systematic numerical approximation of Maxwell's equations in dispersive media. Two discretization strategies are considered, one based on a traditional leapfrog time integration method and the other based on convolution…

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different…

We study the hyperbolic approximation of the Benjamin-Bona-Mahony (BBM) equation proposed recently by Gavrilyuk and Shyue (2022). We develop asymptotic-preserving numerical methods using implicit-explicit (additive) Runge-Kutta methods that…

In this paper we consider a numerical homogenization technique for curl-curl-problems that is based on the framework of the Localized Orthogonal Decomposition and which was proposed in [D. Gallistl, P. Henning, B. Verf\"urth. SIAM J. Numer.…

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at…