Related papers: Error analysis of Runge--Kutta discontinuous Galer…
We present unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows. Our algorithm is geared toward arbitrarily high order approximations in both space and time,…
In this paper, we present a novel class of high-order Runge--Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes…
We consider the discretization of a class of nonlinear parabolic equations by discontinuous Galerkin time-stepping methods and establish a priori as well as conditional a posteriori error estimates. Our approach is motivated by the error…
The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and…
This paper develops three high-order accurate discontinuous Galerkin (DG) methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac (NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG…
A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed…
In this paper, we study high-order exponential time differencing Runge-Kutta (ETD-RK) discontinuous Galerkin (DG) methods for nonlinear degenerate parabolic equations. This class of equations exhibits hyperbolic behavior in degenerate…
This paper presents a fully discrete numerical scheme for one-dimensional nonlocal wave equations and provides a rigorous theoretical analysis. To facilitate the spatial discretization, we introduce an auxiliary variable analogous to the…
Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit…
In this paper, error analysis is established for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve the Vlasov-Maxwell system. This nonlinear hyperbolic system describes the time evolution of collisionless plasma particles of a…
We investigate two efficient time discretizations for the post-processing technique of discontinuous Galerkin (DG) methods to solve hyperbolic conservation laws. The post-processing technique, which is applied at the final time of the DG…
In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related time-Galerkin discretizations. The primary motivation for…
The Runge--Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the…
In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG (RKDG) schemes for hyperbolic…
An $hp$-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, {\it a priori} $hp$-error estimates…
In [11] and [5], an error estimate of optimal convergence rates and optimal error propagation (optimal^2) was given for the Runge-Kutta discontinuous Galerkin (RKDG) method solving the scalar nonlinear conservation laws in the case of…
We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such…
In this article, we consider discrete schemes for a fractional diffusion equation involving a tempered fractional derivative in time. We present a semi-discrete scheme by using the local discontinuous Galerkin (LDG) discretization in the…
We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static,…
We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time dependent) Dirichlet…