Related papers: The new discontinuous Galerkin methods based numer…
The astrophysics of compact objects, which requires Einstein's theory of general relativity for understanding phenomena such as black holes and neutron stars, is attracting increasing attention. In general relativity, gravity is governed by…
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence…
We present and analyse a space-time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential…
This paper, as the sequel to previous work, develops numerical schemes for fractional diffusion equations on a two-dimensional finite domain with triangular meshes. We adopt the nodal discontinuous Galerkin methods for the full spatial…
We propose a high order discontinuous Galerkin (DG) scheme with subcell finite volume (FV) limiter to solve a monolithic first--order hyperbolic BSSNOK formulation of the coupled Einstein--Euler equations. The numerical scheme runs with…
We study the general dynamics of the spherically symmetric gravitational collapse of a massless scalar field. We apply the Galerkin projection method to transform a system of partial differential equations into a set of ordinary…
Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities…
Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by…
Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to…
In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with…
In this study, we propose a unified, general framework for the direct discontinuous Galerkin methods. In the new framework, the antiderivative of the nonlinear diffusion matrix is not needed. This allows a simple definition of the numerical…
I review recent developments in numerical relativity, focussing on progress made in 3D black hole evolution. Progress in development of black hole initial data, apparent horizon boundary conditions, adaptive mesh refinement, and…
We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time…
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…
We present a computational framework for solving the equations of inviscid gas dynamics using structured grids with embedded geometries. The novelty of the proposed approach is the use of high-order discontinuous Galerkin (dG) schemes and a…
Neutron interactions in a fusion power plant play a pivotal role in determining critical design parameters such as coil-plasma distance and breeding blanket composition. Fast predictive neutronic capabilities are therefore crucial for an…
We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy…
We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite…
This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…
A numerical method based on the hybridizable discontinuous Galerkin method in space and backward Euler in time is formulated and analyzed for solving the miscible displacement problem. Under low regularity assumptions, convergence is…