Local Discontinuous Galerkin Method Research Articles

Fourier Analysis of Local Discontinuous Galerkin Methods for Linear Parabolic Equations on Overlapping Meshes

A new local discontinuous Galerkin method for convection–diffusion equations on overlapping mesh was introduced in Du et al. (BIT Numer Math 1–24, 2019). In the new method, the primary variable u and auxiliary variable $$p=u_x$$ are solved on different meshes. The stability and suboptimal error estimates for problems with periodic boundary conditions were derived. Numerical experiments demonstrated that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. Several alternatives to gain optimal convergence rates were demonstrated in Du et al. (2019). However, the reason for accuracy degeneration is still unclear. In this paper, we will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. We explicitly write out the error between the numerical and exact solutions, and investigate the reason for the accuracy degeneration. Moreover, we also find out some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh. Since the current work is based on Fourier analysis, we only consider uniform meshes. Numerical experiments will be given to verify the theoretical analysis.

Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

PurposeThe current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.Design/methodology/approachAt first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.FindingsThe results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.Originality/valueThe local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

Conservative Local Discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction

The formulation of the Local Discontinuous Galerkin (LDG) method applied to the space fractional Klein-Gordon-Schrodinger system with generalized interaction is presented. By considering its primal formulation and taking advantage of the symmetry of the bilinear form associated to the discretization of the Riesz differential operator, conservation of discrete analogues of the mass and the energy can be demonstrated for the semi-discrete problem and for the fully discrete problem using, as time marching scheme, a combination of the modified Crank-Nicolson method for the fractional nonlinear Schrodinger equation and the Newmark method for the nonlinear Klein-Gordon equation. Boundedness of the numerical solution in the L2 norm is derived from the conservation properties of the fully discrete method. A series of numerical experiments with high order approximations illustrates our conservation results and shows that optimal rates of convergence can be also achieved.

Stability analysis and error estimates of local discontinuous Galerkin methods for convection–diffusion equations on overlapping meshes

Local discontinuous Galerkin (LDG) methods are popular for convection–diffusion equations. In LDG methods, we introduce an auxiliary variable p to represent the derivative of the primary variable u, and solve them on the same mesh. In this paper, we will introduce a new LDG method, and solve u and p on different meshes. The stability and error estimates will be investigated. The new algorithm is more flexible and flux-free for pure diffusion equations without introducing additional computational cost compared with the original LDG methods, since it is not necessary to solve each equation twice. Moreover, it is possible to construct third-order maximum-principle-preserving schemes based on the new algorithm. However, one cannot anticipate optimal accuracy in some special cases. In this paper, we will find out the reason for accuracy degeneration which further leads to several alternatives to obtain optimal convergence rates. Finally, several numerical experiments will be given to demonstrate the stability and optimal accuracy of the new algorithm.

Local discontinuous Galerkin methods with implicit-explicit multistep time-marching for solving the nonlinear Cahn-Hilliard equation

In this paper, we develop the fully discrete local discontinuous Galerkin (LDG) methods coupled with the implicit-explicit (IMEX) multistep time marching for solving the nonlinear Cahn-Hilliard equation. To be more specific, we rewrite the Cahn-Hilliard equation in a novel form by adding and subtracting a “linear” term. Then we discretize the spatial derivatives with the LDG methods and the temporal derivative with the IMEX multistep method. Finally, a series of numerical experiments are given to verify the accuracy, efficiency and validness of proposed method.

Optimal stabilization and time step constraints for the forward Euler-Local Discontinuous Galerkin method applied to fractional diffusion equations

A time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order 1<α<2 is considered. By penalyzing the primary variable of the minimal dissipation Local Discontinuous Galerkin (mdLDG) method with a term of order h1−α and using a von Neumann analysis, stability conditions proportional to hα are derived for the forward Euler method and both fractional operators in one dimensional domains. The CFL condition is numerically studied with respect to the approximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when α is close to one. A series of numerical experiments in 1D and 2D problems are presented to validate our theoretical results and those not covered by the theory.

A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form [Formula: see text]. We prove optimal [Formula: see text] error estimates for the solution and for the three auxiliary variables that approximate the first, second, and third-order derivatives. The order of convergence is proved to be [Formula: see text], when piecewise polynomials of degree at most [Formula: see text] are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order [Formula: see text] toward the derivatives of Gauss–Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order [Formula: see text] toward Gauss–Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is [Formula: see text]. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree [Formula: see text] and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

LDG Methods for Reaction-diffusion Systems with Application of Krylov Implicit Integration Factor Methods

In this paper, we present an efficient fully-discrete local discontinuous Galerkin (LDG) method for nonlinear reaction-diffusion systems, which are often used as mathematical models for many physical and biological applications. We can derive numerical approximations not only for solutions but also for their gradients at the same time, while most of methods derive numerical solutions only. And due to the strict time-step restriction ($\Delta t = O(h^2_{\min})$) of explicit schemes for stability, we introduce the implicit integration factor (IIF) method based on Krylov subspace approximation, in which the time step can be taken as $\Delta t = O(h_{\min})$. Moreover, the method allows us to compute element by element and avoid solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical experiments about the reaction-diffusion equations with exact solutions and the well-studied Schnakenberg system are conducted to illustrate the accuracy, capability and advantages of the method.

Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper, we discuss the local discontinuous Galerkin methodscoupled with two specific explicit-implicit-null time discretizations forsolving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$.The basic idea is to add and subtract two equal terms $a_0~U_{xx}$on the right-hand side of the partial differential equation,then to treat the term $a_0~U_{xx}$ implicitly and the other terms$(a(U)U_x)_x-a_0~U_{xx}$ explicitly. We give stability analysis for themethod on a simplified model by the aid of energy analysis,which gives a guidance for the choice of $a_0$, i.e., $a_0~\ge~\max\{a(u)\}/2$to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model, andnumerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.

Local discontinuous Galerkin method for far-field underwater explosion shock wave and cavitation

The shock wave and cavitation are main effects in the far-field underwater explosion, which could cause serious damage to marine structures. In this paper, the fluid mechanical behavior of blast load is described by the propagation of pressure wave. The acoustic pressure caused by far-field explosion is determined by solving the wave equation, where a strongly discontinuous axisymmetric numerical model is established with the local discontinuous Galerkin (LDG) method. The model can calculate the dynamic pressure in the fluid field and capture the high-resolution shock wave. The pressure cutoff model is employed to deal with the cavitation effect due to the reflection of the shock wave. The numerical model is verified by comparing with the analytical solution of the cavitation effect near the structure in one dimension. With the same mesh discretization, the present model shows higher precision than the results calculated by the acoustic finite element method. In addition, the propagation of shock wave in the cylindrical water column is studied. Finally, the formation, growth and collapse of the cavitation region near the free surface are simulated. The LDG model can remove the spurious oscillations behind the shock front and it’s more accurate than the results of the acoustic finite element method, in terms of capturing the sharpness of shock wave and calculating the shock and cavitation loading. And the present model can be applied to calculate the structural damage caused by shock wave in three dimensions.

Two-dimensional simulations and experiments of homogenization and Kirkendall effect in single-phase diffusion triples

A fully discrete local discontinuous Galerkin (LDG) method has been successfully adopted to simulate two-dimensional (2D) homogenization in single-phase diffusion triples controlled by diffusion process. Two Co-Ni and Co-Fe-Ni diffusion triples were prepared and analyzed to verify the simulation results at 1473 K. In areas far away from the triple conjunction of a diffusion triple, there exists composition gradient in only one direction, which can be efficiently predicted by one-dimensional (1D) sharp-interface diffusion simulations (for example, DICTRA: DIffusion-Controlled TRAnsformation simulation software). It has been extensively adopted in the study of diffusional phase transformations readily supported by multi-component multi-phase thermodynamics and full diffusion matrix establishing an integrated thermodynamics-kinetics simulation platform for engineering materials and industrial applications. However, there are composition gradients in two independent directions in the vicinity of triple conjunction (i.e. triple diffusion zone), which cannot be handled on this platform. In the present work, the LDG method combining with alloy thermodynamics and diffusion kinetics was demonstrated to tackle this 2D homogenization problem very well and the effectiveness of this method has been proved from the aspect of numerical analysis by realizing it with a standalone C++ code. In addition to the composition and flux evolutions during 2D homogenization processes, one can reasonably simulate the shift of the Kirkendall plane in the 2D single-phase diffusion zone. This standalone code serves as a solid basis to further consider 2D diffusional multi-phase transformations assuming sharp interfaces in between.

High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative

In this article, a local discontinuous Galerkin (LDG) method is studied for numerically solving the fractal mobile/immobile transport equation with a new time Caputo–Fabrizio fractional derivative. The stability of the LDG scheme is proven, and a priori error estimates with the second‐order temporal convergence rate and the (k + 1)th order spatial convergence rate are derived in detail. Finally, numerical experiments based on Pk, k = 0, 1, 2, 3, elements are provided to verify our theoretical results.

Central local discontinuous Galerkin method for the space fractional diffusion equation

This paper provides the semi-discrete scheme by the central local discontinuous Galerkin method for space fractional diffusion equation on two sets of overlapping cells, and then the stability analysis and error estimates of the scheme are presented. Lastly, we verify the effectiveness of the proposed scheme by the one- and two-dimensional numerical experiments.

Error estimates and post-processing of local discontinuous Galerkin method for Schrödinger equations

In this paper, we present L 2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrödinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrödinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k + 1 , whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2 k . Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O ( h k + 1 ) to at least O ( h 2 k + 1 ) for Schrödinger equations by using alternating numerical fluxes.

Local discontinuous Galerkin method for modeling the nonplanar structures (solitons and shocks) in an electronegative plasma

Using the hydrodynamic equations of cold inertial positive ions with the Maxwellian distribution for light negative ion and electron densities and the Poisson equation, the family of nonplanar (cylindrical and spherical) Korteweg-de Vries (KdV) equations, i.e., the KdV, modified KdV, and extended KdV (EKdV), are obtained for small but finite amplitude ion-acoustic waves. The nonplanar EKdV equation is used to analyze the time-dependent planar and nonplanar soliton and shock structures. It is well-known that the exact solutions of the family of nonplanar KdV equations are not possible. Therefore, a local discontinuous Galerkin (LDG) method is developed for solving the nonplanar EKdV equation numerically. According to this method, the initial and boundary conditions for the solitary and shock waves are accurately identified. The L2 stability of the LDG method is proved for the general nonlinear case. The existence regions for both solitary and shock excitations have been defined precisely depending on the laboratory plasma parameters. Moreover, the effects of the negative-ion temperature ratio and the negative ion concentration on the profiles of the nonlinear structures (solitons and shocks) are examined. The effect of the geometrical divergence on the pulse profile is also reported which indicates that the localized pulses deform as time goes on. Furthermore, it is found that the amplitude of cylindrical structures (solitons and shocks) is larger than that of planar ones but smaller than that of the spherical ones. Moreover, in cylindrical geometry, the nonlinear structures travel slower than in the spherical ones. The implications of our results agree with the experimental observations.

Superconvergence analysis of local discontinuous Galerkin methods for linear convection–diffusion equations in one space dimension

This paper is concerned with the superconvergence study of the local discontinuous Galerkin (LDG) method for one-dimensional time-dependent linear convection–diffusion equations, where the convection flux is taken as the upwind flux, while the diffusion fluxes chosen as the alternating fluxes. Superconvergence properties for both the solution itself and auxiliary variables are established. Precisely, we prove that, the LDG solutions are superconvergent with an order of \(k+2\) towards a particular projection of the exact solution and the auxiliary variable, and thus a \(k+1\)-th order superconvergence for the derivative approximation and a \(k+2\)-th order superconvergence for the function value approximation at a class of Radau points are obtained. Especially, we show that the convergence rate of the derivative approximation for the exact solution can reach \(k+2\) when the convection flux is the same as the diffusion flux, two order higher than the optimal convergence rate. Furthermore, a \(2k+1\)-th order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages, is also obtained under some suitable initial discretization. Numerical experiments indicate that most of our theoretical findings are optimal.

A local discontinuous Galerkin method for the compressible Reynolds lubrication equation

We present an extension of the local discontinuous Galerkin (LDG) method introduced in [9] for nonlinear diffusion problems to nonlinear stationary convection–diffusion problems. We develop a numerical study of the convergence properties of the new method and solve the stationary compressible Reynolds lubrication equation under some realistic conditions.

Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems

We formulate a coupled surface/subsurface flow model that relies on hydrostatic equations with free surface in the free flow domain and on Darcy’s law in the subsurface. The model is discretized using the local discontinuous Galerkin method, and a statement of discrete energy stability is proved for the fully non-linear coupled system. The performance of the proposed scheme is illustrated by a numerical convergence study and a realistic example.

Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients

Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities of the media. This work considers a two-dimensional wave equation with random coefficients which may be discontinuous in space. Generalized polynomial chaos method is used in conjunction with stochastic Galerkin approximation, and local discontinuous Galerkin method is used for spatial discretization. Our method is shown to be energy preserving in semi-discrete form as well as in fully discrete form, when leap-frog time discretization is used. Its convergence rate is proved to be optimal and the error grows linearly in time. The theoretical properties of the proposed scheme are validated by numerical tests.