Recent work on solving partial differential equations (PDEs) with deep neural networks
(DNNs) is presented. The paper reviews and extends some of these methods while carefully …

Recent work has introduced a simple numerical method for solving partial differential
equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the …

This paper presents a shock capturing artificial viscosity scheme for the compact high-order
finite volume methods in terms of the variational reconstructions on unstructured grids. The …

We introduce a novel positivity-preserving numerical stabilisation approach for high-order
discontinuous spectral element approximations of compressible multi-species flows. The …

In this paper, we develop an oscillation-free discontinuous Galerkin (OFDG) method for
solving the shallow water equations with a non-flat bottom topography. Due to the nonlinear …

In the modelling of hydrodynamics, the Discontinuous Galerkin (DG) approach constitutes a
more complex and modern alternative to the well-established finite volume method. The …

The high-order numerical solution of the non-linear shallow water equations is susceptible
to Gibbs oscillations in the proximity of strong gradients. In this paper, we tackle this issue by …

A high‐order‐accurate weighted essentially non‐oscillatory (WENO) limited upwind finite‐
volume scheme is detailed for the compressible, nonhydrostatic, inviscid Euler equations …

GPU accelerated high order reconstruction of signed distance function of the level set
method is studied. The flow based reinitialization equation is discretized in space by using a …

The nodal discontinuous Galerkin (DG) methods possess many good properties that make
them very attractive for numerically solving the shallow water equations, but it is necessary …