Local Discontinuous Galerkin Scheme Research Articles

Boundary Treatment for High-Order IMEX Runge–Kutta Local Discontinuous Galerkin Schemes for Multidimensional Nonlinear Parabolic PDEs

Boundary Treatment for High-Order IMEX Runge–Kutta Local Discontinuous Galerkin Schemes for Multidimensional Nonlinear Parabolic PDEs

Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

Numerical Studies on the Classical Long Wave System Describing Shallow Water Waves With Dispersion

In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution.

Analysis and application of a local discontinuous Galerkin method for the electromagnetic concentrator model

AbstractIn this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi‐discrete and full‐discrete LDG schemes. Numerical results are presented to justify the theoretical analysis and demonstrate the interesting wave concentration property by the electromagnetic concentrator.

Local discontinuous Galerkin method for a nonlocal viscous water wave model

In this article, the fully discrete local discontinuous Galerkin (LDG) method is presented for numerically solving a class of nonlocal viscous water wave model, where the BDF2 with the L1 formula of the time Caputo fractional derivative is applied to deal with the time direction, and the LDG method is developed to approximate the space direction. The stability of the fully discrete LDG scheme is proven, and the a priori error result with O(τ32+hk+12) in L2-norm is derived in detail. Finally, numerical examples supporting theoretical error result are provided, and the decay rates under different coefficients are also discussed.

Local discontinuous Galerkin method for the Riesz space distributed-order Sobolev equation

We present and analyze a local discontinuous Galerkin finite element method to solve the Riesz space distributed-order Sobolev equation. To approximate the distributed-order Riesz space derivative, the Gauss quadrature as the high computational accuracy method is proposed. A multi-term fractional equation is then constructed from the considered equation by approximating the Riesz space derivative. Moreover, stability analysis is provided for a semi-discrete scheme. We provide numerical results to justify the theoretical analysis by solving the ordinary differential equation. We obtain this after implementing the local discontinuous Galerkin scheme on the Riesz space distributed-order Sobolev equation with the Crank–Nicolson scheme as a time marching method and the Laplace transform technique.

Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation

The goal of this article is to develop local discontinuous Galerkin (LDG) schemes for solving a time fractional equation describing the ultrasonic wave in a homogeneous isotropic porous material. Two novel semi-discrete LDG schemes are designed for the considered model. The semi-discrete LDG schemes are constructed by splitting the original model into a coupled system. The first semi-discrete scheme follows the traditional LDG method by splitting second-order space derivative. The second one splits the original model for both time and space derivatives. The discontinuous Galerkin is used for the spatial discretization. Two kinds of fully discrete LDG schemes are presented by using the Grünwald-Letnikov and L1 approximation formulas for the time fractional derivatives. The $ L^2 $ norm stability and convergence analysis are carried out for both semi-discrete and fully discrete LDG schemes. The stability analysis reveals that the numerical schemes are unconditionally stable in $ L^2 $ norm and convergence with optimal convergence rate. Finally, numerical examples are presented to test the effectiveness of the proposed schemes and the correctness of the theoretical analysis.

A fully discrete local discontinuous Galerkin method for variable-order fourth-order equation with Caputo-Fabrizio derivative based on generalized numerical fluxes

<abstract><p>In this paper, an effective numerical method for the variable-order(VO) fourth-order problem with Caputo-Fabrizio derivative will be constructed and analyzed. Based on generalized alternating numerical flux, appropriate spatial and temporal discretization, we get a fully discrete local discontinuous Galerkin(LDG) scheme. The theoretic properties of the fully discrete LDG scheme are proved in detail by mathematical induction, and the method is proved to be unconditionally stable and convergent with $ {\rm O}(\tau+{h^{k+1}}) $, where $ h $ is the spatial step, $ \tau $ is the temporal step and $ k $ is the degree of the piecewise $ P^k $ polynomial. In order to show the efficiency of our method, some numerical examples are carried out by Matlab.</p></abstract>

A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative

<abstract><p>In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For $ 0 < \alpha(t) < 1 $, we prove that the method is unconditionally stable and the errors attain $ (k+1) $-th order of accuracy for piecewise $ P^k $ polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.</p></abstract>

Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k≥2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ=O(hk) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method discussed in this paper is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss the high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations, respectively. By the aid of the Fourier method, we perform stability analysis for the schemes on the simplified equations with periodic boundary conditions, which demonstrates the stability criteria for the resulting schemes. Even though the analysis is only performed on the simplified equations, numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations.

Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian (−Δ)s with s∈(0,1) in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than O(hk+12) with k≥1.

Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

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

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 Methods to a Dispersive System of KdV-Type Equations

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the dispersive system of Korteweg–de Vries (KdV) type equations. Based on a cardinal conservative quantity of this system, we design and discuss two different types of numerical fluxes, including the conservative and dissipative ones for the linear and nonlinear terms respectively. Thus, one conservative together with three dissipative LDG schemes for the KdV-type system are developed in our paper. The invariant preserving property for the conservative scheme and corresponding dissipative properties for the other three dissipative schemes are all presented and proven in this paper. The error estimates for two schemes are given, whose numerical fluxes for linear terms are chosen as the dissipative type. Assuming that the discontinuous piecewise polynomials of degree less than or equal to k are adopted, and conservative numerical fluxes are employed to discretize the nonlinear terms, we obtain a suboptimal a priori bound of order k; yet in the case of dissipative fluxes, we obtain a slightly better bound of order $$k+\frac{1}{2}$$ . Numerical experiments for this system in different circumstances are provided, including accuracy tests for two kinds of traveling waves, long-time simulations for solitary waves and interactions of multi-solitary waves, to illustrate the accuracy and capability of these schemes.

Local discontinuous Galerkin method for time variable order fractional differential equations with sub-diffusion and super-diffusion

This paper presents an approximate solution of time variable order fractional differential equations with sub-diffusion and super-diffusion. The aim of paper is to solve and analyze this problem by a fully discrete local discontinuous Galerkin scheme. The method is based on local discontinuous Galerkin method in space and a finite difference technique in time. The numerical stability and convergence of the proposed method are investigated then the convergence rate O(hk+1+△t2−α(tn)) in the case of sub-diffusion and O(hk+1+△t) in the case of super-diffusion are proven for the presented scheme. Finally, provided numerical examples illustrate efficiency of the method and accuracy of the theory.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order $$ \theta $$ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional diffusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with $$O(h^{k+1}+\varDelta t^2)$$ is derived, where $$k\geqslant 0$$ denotes the index of the basis function. Extensive numerical results with $$Q^k(k=0,1,2,3)$$ elements are provided to confirm our theoretical results, which also show that the second-order convergence rate in time is not impacted by the changed parameter $$\theta$$ .

Efficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methods

Author(s): Saye, RI | Abstract: With an emphasis on achieving ideal multigrid solver performance, this paper explores the design of local discontinuous Galerkin schemes for multiphase elliptic interface problems. In particular, for cases exhibiting coefficient discontinuities several orders in magnitude, the role of viscosity-weighted numerical fluxes on interfacial mesh faces is examined: findings support a known strategy of harmonic weighting, but also show that further improvements can be made via a stronger kind of biasing, denoted herein as viscosity-upwinded weighting. Applying this strategy, multigrid performance is assessed for a variety of elliptic interface problems in 1D, 2D, and 3D, across 16 orders of viscosity ratio. These include constant-and variable-coefficient problems, multiphase checkerboard patterns, implicitly defined interfaces, and 3D problems with intricate geometry. With the exception of a challenging case involving a lattice of vanishingly small droplets, in all demonstrated examples the condition number of the multigrid V-cycle preconditioned system has unit order magnitude, independent of the mesh size h.

LDG schemes with second order implicit time discretization for a fractional sub-diffusion equation

In this paper, we propose new local discontinuous Galerkin (LDG) schemes for solving a time fractional sub-diffusion equation. The new LDG schemes is constructed rely on the splitting of time fractional derivative and space derivative. The key idea of semi-discrete LDG scheme is to split the time fractional derivative into a weak singular integral and a classic first order derivative. By using second time discretization to approximate the weak singular integral and the classic first order derivative, two fully discrete LDG schemes with second order implicit time discretization for the time fractional diffusion equation are presented. The LDG finite element approximations is used in space variable, and Crank–Nicholson, second order backward differentiation formulas are used for the temporal approximation. The error estimates of the presented semi-discrete and fully-discrete LDG schemes are both analyzed. The analysis results show that our numerical schemes are unconditional stable with the second order accuracy in temporal discretization. Numerical experiments are presented to confirm the theoretical results.

Steady-State Simulation of Semiconductor Devices Using Discontinuous Galerkin Methods

Design of modern nanostructured semiconductor devices often calls for simulation tools capable of modeling arbitrarily-shaped multiscale geometries. In this work, to this end, a discontinuous Galerkin (DG) method-based framework is developed to simulate steady-state response of semiconductor devices. The proposed framework solves a system of Poisson equation (in electric potential) and stationary drift-diffusion equations (in charge densities), which are nonlinearly coupled via the drift current and the charge distribution. This system is “decoupled” and “linearized” using the Gummel method and the resulting equations are discretized using a local DG scheme. The proposed framework is used to simulate geometrically intricate semiconductor devices with realistic models of mobility and recombination rate. Its accuracy is demonstrated by comparing the results to those obtained by the finite volume and finite element methods implemented in a commercial software package.

Local discontinuous Galerkin methods for the time tempered fractional diffusion equation

In this article, we consider high order discrete schemes for solving the time tempered fractional diffusion equation. We present a semi-discrete scheme by using the local discontinuous Galerkin (LDG) discretization in the spatial variable. We prove that the semi-discrete scheme is unconditionally stable in L2 norm and convergence with optimal convergence rate O(hk+1), where h is the spatial step size. We develop a class of fully discrete LDG schemes by combining the weighted and shifted Lubich difference operators with respect to the time variable, and establish the error estimates. Finally, numerical experiments are presented to verify the theoretical results.