Discrete Local Discontinuous Galerkin Research Articles

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

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.

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>

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>

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.

Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation

<abstract><p>In this paper, a fully discrete local discontinuous Galerkin finite element method is proposed to solve the KdV-Burgers-Kuramoto equation with variable-order Riemann-Liouville time fractional derivative. The method proposed in this paper is based on the finite difference method in time and local discontinuous Galerkin method in space. For all $ \epsilon(t)\in (0, 1) $ with variable order, we prove the scheme is unconditional stable and convergent. Finally, numerical examples are provided to verify the theoretical analysis and the order of convergence for the proposed method.</p></abstract>

Error estimates of local discontinuous Galerkin method with implicit-explicit Runge Kutta for two-phase miscible flow in porous media

Two-phase miscible flow in porous media is described by a coupled system of nonlinear partial differential equations. In this paper, we develop and analyze the fully discrete local discontinuous Galerkin method coupled with the implicit-explicit Runge Kutta (IMEX-RK) method for the problems. Second-order time discretization scheme in two adjacent time layer can be obtained with the IMEX-RK method. For the temporal-spatial discretization schemes, we give the optimal convergence analysis for both the pressure and concentration in L2-norm. We present several numerical examples for two-phase miscible flow problem to demonstrate the effectiveness and robustness of our method.

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.

Energy stable discontinuous Galerkin method for compressible Navier–Stokes–Allen–Cahn system

In this paper, we present a fully discrete local discontinuous Galerkin (LDG) finite element method combined with scalar auxiliary variable (SAV) approach for the compressible Navier–Stokes–Allen–Cahn (NSAC) system. We start with a linear and first order scheme for time discretization and the minimal dissipation LDG for spatial discretization, which is based on the SAV approach and is proved to be unconditionally energy stable for one dimensional case. The velocity, the density and the mass concentration of fluid mixture can be solved separately. In addition, a semi-implicit spectral deferred correction (SDC) method combined with the first order scheme is employed to improve the temporal accuracy. Due to the local properties of the LDG methods, the resulting algebraic equations at the implicit level are easy to implement. In particular, we use efficient and practical multigrid solvers to solve the resulting algebraic equations. Although there is no proof of stability for the semi-implicit SDC with LDG spatial discretization, numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the discretized energy stablity, the capability and efficiency of the proposed method. Numerical results show that the initial state determines the long time behavior of the diffusive interface for the two–phase flow, which are consistent with theoretical asymptotic stability results in Chen et al. (2018)[1].

A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order \begin{document}$ O(h^{k+1}+\tau^{2-\alpha}) $\end{document} , where \begin{document}$ h, \tau $\end{document} and \begin{document}$ k $\end{document} are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grunwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the $$L^2$$ -convergence of the scheme are proved for all variable-order $$\alpha (t)\in (0,1)$$ . The proposed method is of accuracy-order $$O(\tau ^3+h^{k+1})$$ , where $$\tau$$ , h, and k are the temporal step size, the spatial step size, and the degree of piecewise $$P^k$$ polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

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.

Stability analysis and error estimates of local discontinuous Galerkin methods with semi-implicit spectral deferred correction time-marching for the Allen–Cahn equation

This paper is concerned with the stability and error estimates of the local discontinuous Galerkin (LDG) method coupled with semi-implicit spectral deferred correction (SDC) time-marching up to third order accuracy for the Allen–Cahn equation. Since the SDC method is based on the first order convex splitting scheme, the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step, which increases the difficulty of the theoretical analysis. For the LDG discretizations coupled with the second and third order SDC methods, we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces. At the same time, the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis. Comparing to the Runge–Kutta type semi-implicit schemes which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. We provide two different ideas to overcome the difficulty of the nonlinear terms. By choosing the test functions carefully, the energy stability and error estimates are obtained in the sense that the time step Δt only requires a positive upper bound and is independent of the mesh size h. Numerical examples are presented to illustrate our theoretical results.

A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient

In this paper, a fully discrete Local discontinuous Galerkin (LDG) method is discussed for a time-fractional reaction diffusion initial-boundary value problem with discontinuous diffusive coefficient. In this method the well-known L2-1σ scheme on graded meshes is presented to deal with the weak singularity at initial time t=0, while in the spatial direction a LDG method on a uniform mesh is presented to handle the discontinuous coefficient. By the discrete fractional Gronwall inequality, the L2-norm stability and consistency estimate results are derived for the proposed fully discrete LDG method. Numerical experiments are presented to verify the sharpness of the error analysis.

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.

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.

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.

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.

Energy Dissipative Local Discontinuous Galerkin Methods for Keller–Segel Chemotaxis Model

In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller–Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.

Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge‐Kutta time‐marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge‐Kutta–LDG schemes by energy analysis, under the condition τ ≤ λh2, where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu‐Ni system, the numerical results show good agreement with the experimental results.