Discrete Local Discontinuous Galerkin Research Articles

Error Estimate on a Fully Discrete Local Discontinuous Galerkin Method for Linear Convection-Diffusion Problem

In this paper we present the error estimate for the fully discrete local discontinuous Galerkin algorithm to solve the linear convection-diffusion equation with Dirichlet boundary condition in one dimension. The time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the reasonable temporal-spatial condition as general. The optimal error estimate in both space and time is obtained by aid of the energy technique, if we set the numerical flux and the intermediate boundary condition properly.

Analysis of a local discontinuous Galerkin method for time‐fractional advection‐diffusion equations

PurposeThe purpose of this paper is to develop a fully discrete local discontinuous Galerkin (LDG) finite element method for solving a time‐fractional advection‐diffusion equation.Design/methodology/approachThe method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space.FindingsBy choosing the numerical fluxes carefully the authors' scheme is proved to be unconditionally stable and gets L2 error estimates of O(hk+1+(Δt)2+(Δt)α/2hk+(1/2)). Finally Numerical examples are performed to illustrate the effectiveness and the accuracy of the method.Originality/valueThe proposed method is different from the traditional LDG method, which discretes an equation in spatial direction and couples an ordinary differential equation (ODE) solver, such as Runger‐Kutta method. This fully discrete scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. Numerical examples prove that the authors' method is very effective. The present paper is the authors' first step towards an effective approach based on the discontinuous Galerkin method for the solution of fractional‐order problems.

Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation

This paper aims to develop a fully discrete local discontinuous Galerkin finite element method for numerical simulation of the time-fractional telegraph equation, where the fractional derivative is in the sense of Caputo. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. The stability and convergence of this discontinuous approach are discussed and theoretically proven. Finally numerical examples are performed to illustrate the effectiveness and the accuracy of the method.

Energy conserving local discontinuous Galerkin methods for wave propagation problems

Wave propagation problems arise in a wide range of applications. The energyconserving property is one of the guiding principles fornumerical algorithms, in orderto minimize the phase or shape errors after long time integration.In this paper, we develop and analyze a local discontinuous Galerkin (LDG)method for solving the wave equation. We prove optimal error estimates,superconvergence toward a particular projection of the exact solution,and the energy conserving property for the semi-discrete formulation.The analysis is extended to the fully discrete LDG scheme,with the centered second-order time discretization(the leap-frog scheme).Our numerical experiments demonstrate optimal rates of convergenceand superconvergence.We also show that the shape of the solution, after long timeintegration, is well preserved due to the energy conserving property.

Analysis of the fractional Kawahara equation using an implicit fully discrete local discontinuous Galerkin method

AbstractIn this article, an implicit fully discrete local discontinuous Galerkin (LDG) finite element method, on the basis of finite difference method in time and LDG method in space, is applied to solve the time‐fractional Kawahara equation, which is introduced by replacing the integer‐order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and convergent through analysis. Extensive numerical results are provided to demonstrate the performance of the present method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

NUMERICAL ANALYSIS OF AN IMPLICIT FULLY DISCRETE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE FRACTIONAL ZAKHAROV–KUZNETSOV EQUATION

In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Zakharov–Kuznetsov equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditional stable and L 2 error estimate for the linear case with the convergence rate through analysis.

A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system

In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for solving the time-fractional coupled Schrödinger system. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Through analysis we show that our scheme is unconditionally stable, and the L2 error estimate has the convergence rate O(hk+1+(Δt)2+(Δt)α2hk+12) for the linear case. Extensive numerical results are provided to demonstrate the efficiency and accuracy of the scheme.

Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time‐fractional KdV‐Burgers‐Kuramoto equation

AbstractIn this paper, an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time‐fractional KdV‐Burgers‐Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and L2 error estimate for the linear case with the convergence rate O(hk+1 + (Δ t)2+ (Δ t)α/2hk+$^{1 \over 2}$). Numerical examples are presented to show the efficiency and accuracy of our scheme.

Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation

In this paper we present and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for solving the time-fractional Schrödinger equation, where the fractional derivative is described in the Caputo sense. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. A stability and error analysis is performed on the numerical methods. Numerical results confirm the expected convergence rates and illustrate the effectiveness of the method.

Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations

In this paper, we introduce a general approach for proving optimal $L^2$ error estimates for the semidiscrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates holds not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multidimensional Schrodinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization by using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy.

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms: Formulation and Analysis for Third Derivative Spatial Terms Using the LDG Discretization Procedure

AbstractIn this paper, we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume (SV) context; more specifically the emphasis is on handling equations containing third derivative terms. This formulation is based on the LDG (Local Discontinuous Galerkin) flux discretization method, originally employed for viscous equations containing second derivatives. A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation. The Fourier analysis was utilized for two purposes: firstly to eliminate all the unstable SV partitions, secondly to obtain the optimal SV partition. Numerical experiments are performed to illustrate the capability of this formulation. Since this formulation is extremely local, it can be easily parallelized and a h-p adaptation is relatively straightforward to implement. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries (KdV) type problems.

The role of the penalty in the local discontinuous Galerkin method for Maxwell’s eigenvalue problem

The local discontinuous Galerkin (LDG) method is a promising emerging technology for discretizing a wide range of differential operators. It is well known that the robustness of this method is contingent upon a stabilization term. We analyze the role of this term in the LDG discretization of the Maxwell curl–curl operator, demonstrating that the stabilization parameter can be used to split the spectrum into two sections. The first set of eigenvalues approximate the spectrum of the curl-conforming finite element curl–curl operator, while the second set are spurious eigenvalues that can be made arbitrarily large by increasing the stabilization parameter. We confirm this analysis with a range of computational experiments that compare the spectra obtained from LDG and curl-conforming discretizations.