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We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the…
We consider the initial/boundary value problem for the fractional diffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two "simple" fully discrete schemes based on the Galerkin finite element…
We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data…
In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for time-dependent transport equations in multi-dimensions. The method is constructed using multiwavlelets on tensorized nested grids. Adaptivity is…
Numerical climate- and weather-prediction requires the fast solution of the equations of fluid dynamics. Discontinuous Galerkin (DG) discretisations have several advantageous properties. They can be used for arbitrary domains and support a…
The scalability of time-dependent partial differential equation (PDE) solvers based on the discontinuous Galerkin (DG) method is increasingly limited by data communication and synchronization requirements across processing elements (PEs) at…
We theoretically analyze the superconvergence of the upwind discontinuous Galerkin (DG) method for both the steady-state and time-dependent radiative transfer equation (RTE), and apply the Smooth-Increasing Accuracy-Conserving (SIAC)…
A hybridized discontinuous Galerkin method is proposed for solving 2D fractional convection-diffusion equations containing derivatives of fractional order in space on a finite domain. The Riemann-Liouville derivative is used for the spatial…
The balancing domain decomposition methods (BDDC) are originally introduced for symmetric positive definite systems and have been extended to the nonsymmetric positive definite system from the linear finite element discretization of…
In this paper, a space-time discontinuous Galerkin finite element method for distributed optimal control problems governed by unsteady diffusion-convection-reaction equations with control constraints is studied. Time discretization is…
We propose IMEX HDG-DG schemes for planar and spherical shallow water systems. Of interest is subcritical flow, where the speed of the gravity wave is faster than that of nonlinear advection. In order to simulate these flows efficiently, we…
We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229, 3091-3120) to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for…
A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space…
We present and analyze an a posteriori error estimator for a space-time hybridizable discontinuous Galerkin discretization of the time-dependent advection-diffusion problem. The residual-based error estimator is proven to be reliable and…
This article investigates matrix-free higher-order discontinuous Galerkin discretizations of the Navier--Stokes equations for incompressible flows with variable viscosity. The viscosity field may be prescribed analytically or governed by a…
In this paper, a new strategy for a sub-element based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low to high order…
A diffused-interface approach based on the Allen-Cahn phase field equation is developed within a high-order Discontinuous Galerkin framework. The interface capturing technique is based on the balance between explicit diffusion and…
The paper proposes and analyzes an efficient second-order in time numerical approximation for the Allen-Cahn equation, which is a second order nonlinear equation arising from the phase separation model. We firstly present a fully discrete…
In this paper, by introducing a reconstruction operator based on the Legendre moments, we construct a reduced discontinuous Galerkin (RDG) space that could achieve the same approximation accuracy but using fewer degrees of freedom (DoFs)…
In this paper we present a new high order semi-implicit DG scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible…