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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…
In the present paper, we consider the discontinuous Galerkin (DG) methods for solving short pulse (SP) type equations. The short pulse equation has been shown to be completely integrable, which admits the loop-soliton, cuspon-soliton…
The discontinuous Galerkin dG method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily…
We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin…
This work considers the Galerkin approximation and analysis for a hyperbolic integrodifferential equation, where the non-positive variable-sign kernel and nonlinear-nonlocal damping with both the weak and viscous damping effects are…
In this work, we present a new high order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the…
The purpose of this work is to propose a novel a posteriori finite volume subcell limiter technique for the Discontinuous Galerkin finite element method for nonlinear systems of hyperbolic conservation laws in multiple space dimensions that…
This paper proposes and analyzes a fully discrete scheme that discretizes space with an ultra-weak local discontinuous Galerkin scheme and time with the Crank--Nicolson method for the nonlinear biharmonic Schr\"odinger equation. We first…
In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017) to the…
Variable-order time-fractional wave equations provide a flexible model for wave phenomena with evolving memory effects and anomalous temporal dynamics. Their numerical approximation is challenging because the variable-order fractional…
In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a $hp$-version discontinuous Galerkin finite element approximation in the time…
We consider the iterative solution of anisotropic radiative transfer problems using residual minimization over suitable subspaces. We show convergence of the resulting iteration using Hilbert space norms, which allows us to obtain…
In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and…
This paper presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian-Eulerian (ALE)…
This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG)…
The high-order accurate continuous Galerkin finite element method offers attractive computational efficiency for computational fluid dynamics. A challenge is however spurious oscillations which result for convection dominated flows over…
We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear…
In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and…
In this work, an exponential Discontinuous Galerkin (DG) method is proposed to solve numerically Vlasov type equations. The DG method is used for space discretization which is combined exponential Lawson Runge-Kutta method for time…
An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for…