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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…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
An acoustic wave propagation problem with a log normal random field approximation for wave speed is solved using a sampling-free intrusive stochastic Galerkin approach. The stochastic partial differential equation with the inputs and…
In the framework of ODEs, we uncover a new link between the continuous Galerkin method (see Math. Comp. (1972), 26 (118 and 120), 415-426 and 881-891) and the discontinuous Galerkin method (see Mathematical Aspects of Finite elements in…
Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically…
A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
In this paper two new families of arbitrary high order accurate spectral DG finite element methods are derived on staggered Cartesian grids for the solution of the inc.NS equations in two and three space dimensions. Pressure and velocity…
In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$. The basic idea is to add and…
To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields…
We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on…
Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…
Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to…
This work aims to construct and analyze a discontinuous Galerkin method on polytopal grids (PolydG) to solve the pseudo-stress formulation of the unsteady Stokes problem. The pseudo-stress variable is introduced due to the growing interest…
This article discusses the uncertainty quantification (UQ) for time-independent linear and nonlinear partial differential equation (PDE)-based systems with random model parameters carried out using sampling-free intrusive stochastic…
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…
We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…
We introduce the concept of half-closed nodes for nodal discontinuous Galerkin (DG) discretisations. Unlike more commonly used closed nodes in DG, where on every element nodes are placed on all of its boundaries, half-closed nodes only…
Presented here is a preliminary study of a strictly linear, discontinuous-Petrov-Galerkin scheme for the discrete-ordinates method in slab geometry. By ``linear'', we mean the discretization does not depend on the solution itself as is the…
Standard discontinuous Galerkin methods, based on piecewise polynomials of degree $ \qq=0,1$, are considered for temporal semi-discretization for second order hyperbolic equations. The main goal of this paper is to present a simple and…