A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

2014 ◽  
Vol 15 (4) ◽  
pp. 1091-1107 ◽  
Author(s):  
Yinhua Xia ◽  
Yan Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Time Discretization ◽  
Local Discontinuous Galerkin Method ◽  
Implicit Time ◽  
Discrete Version ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Spatial Derivatives ◽  
Number Of Particles

AbstractIn this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fengna Yan ◽  
Yan Xu
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Implicit Time ◽  
Time Step ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Local Discontinuous Galerkin ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Abstract In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn–Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn–Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L ∞ L^{\infty} -norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.

Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yao Cheng ◽  
Chuanjing Song ◽  
Yanjie Mei
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Domain Boundary ◽  
Local Discontinuous Galerkin Method ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Element Space ◽  
Local Discontinuous Galerkin ◽  
Ldg Method

AbstractLocal discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer. The method is equipped with a general numerical flux including two kinds of independent parameters. By virtue of the weighted estimates and suitably designed global projections, we establish optimal {(k+1)}-th error estimate in a local region at a distance of {\mathcal{O}(h\log(\frac{1}{h}))} from domain boundary. Here k is the degree of piecewise polynomials in the discontinuous finite element space and h is the maximum mesh size. Both semi-discrete LDG method and fully discrete LDG method with a third-order explicit Runge–Kutta time-marching are considered. Numerical experiments support our theoretical results.

scholarly journals A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xuehai Huang
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Local Discontinuous Galerkin Method ◽  
Element Space ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Nearly Incompressible Linear Elasticity

A reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity is proposed in this paper, which is locking-free. RLDG method can be formally regarded as a special case of LDG method withC11=0. However, RLDG method is actually not covered by LDG method, whereC11must be chosen to be positive to ensure the stability of LDG method. RLDG method can also be considered as the localization of some symmetric nonconforming mixed finite element method. The implementation of RLDG method is discussed. By introducing a lifting operator as LDG method, RLDG method can be rewritten as primal formulation with unknown displacement only. Next, we obtain that the convergence rates of the approximation to stress tensor in energy norm and displacement inL2-norm areO(hk)andO(hk+1), respectively, which are both uniform with respect toλ. Moreover, we obtain aH(div)-conforming displacement by projecting the displacement and corresponding numerical trace of RLDG method into the Raviart-Thomas element space. And then we analyze the error estimates of this postprocessed displacement inH(div)-seminorm andL2-norm, which are also uniform with respect toλ. Finally, some numerical results are shown to demonstrate the theoretical results.

scholarly journals A local discontinuous Galerkin method for nonlinear parabolic SPDEs

2020 ◽  
Author(s):  
Yunzhang Li ◽  
Chi-Wang Shu ◽  
Shanjian Tang
Keyword(s):  
Differential Equations ◽  
Discontinuous Galerkin ◽  
Numerical Scheme ◽  
Hyperbolic Equations ◽  
Time Discretization ◽  
Local Discontinuous Galerkin Method ◽  
Optimal Error Estimates ◽  
Fully Nonlinear ◽  
Local Discontinuous Galerkin ◽  
Dg Method

In this paper, we propose a local discontinuous Galerkin (LDG) method for fully nonlinear and possibly degenerate parabolic stochastic partial differential equations (SPDEs), which is a high-order numerical scheme. This method is an extension of the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and share with the DG method its advantage and flexibility. We prove the $L^2$-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ($\cO(h^{k+1})$) for smooth solutions of semilinear stochastic equations is shown if polynomials of degree $k$ are used. We also develop an explicit derivative-free order $1.5$ time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples displaying the performance of the method are shown.

Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations

2015 ◽  
Vol 17 (2) ◽  
pp. 424-457 ◽  
Author(s):  
Ruihan Guo ◽  
Yan Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Equations ◽  
Three Dimensional ◽  
Mode Analysis ◽  
Implicit Time ◽  
Local Mode ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Time Marching ◽  
Kdv Type Equations

AbstractIn this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step , with the k-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

scholarly journals On the Local Discontinuous Galerkin Method for Linear Elasticity

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Yuncheng Chen ◽  
Jianguo Huang ◽  
Xuehai Huang ◽  
Yifeng Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Linear Elasticity ◽  
General Framework ◽  
Local Discontinuous Galerkin Method ◽  
Convergence Theory ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Dg Methods ◽  
Discrete Stability

Following Castillo et al. (2000) and Cockburn (2003), a general framework of constructing discontinuous Galerkin (DG) methods is developed for solving the linear elasticity problem. The numerical traces are determined in view of a discrete stability identity, leading to a class of stable DG methods. A particular method, called the LDG method for linear elasticity, is studied in depth, which can be viewed as an extension of the LDG method discussed by Castillo et al. (2000) and Cockburn (2003). The error bounds inL2-norm,H1-norm, and a certain broken energy norm are obtained. Some numerical results are provided to confirm the convergence theory established.

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