Local Discontinuous Galerkin Research Articles

Local discontinuous Galerkin methods with implicit–explicit BDF time marching for Newell–Whitehead–Segel equations

The Newell–Whitehead–Segel type equations with time-dependent Dirichlet boundary conditions are solved by the local discontinuous Galerkin (LDG) method coupled with the implicit–explicit backward difference formulas (IMEX-BDF). With a suitable setting of numerical fluxes and by the aid of the multiplier technique and the a priori error assumption technique, the optimal error estimate for the corresponding fully discrete LDG-IMEX-BDF schemes is obtained by energy analysis, under the condition τ ≤ C h 1 / s , where h and τ are mesh size and time step, respectively, the positive constant C is independent of h, and s = 1 , … , 5 is the order of the IMEX-BDF method. Numerical experiments are also presented to verify the accuracy of the considered schemes.

Higher order method based on the coupling of local discontinuous Galerkin method with multistep implicit-explicit time-marching for the micropolar Navier–Stokes equations

In this paper, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal multistep implicit-explicit (IMEX) evolution for the micropolar Navier–Stokes equations (MNSE) is adopted to construct an efficient fully discrete method. First, the multistep IMEX-LDG methods are constructed up to the third order, which have small computational scale and facilitate implementation to higher orders. Second, the unconditional stability of the fully discrete method on Cartesian grids is derived theoretically, meaning that the temporal step size τ is upper bounded by a positive constant independent of the spatial mesh size h. Last, numerical results for nonlinear MNSE with different boundary conditions are presented, confirming the theoretical validity and effectiveness of the method.

A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation

A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients

In this research, we develop an innovative and efficient numerical method for solving a nonlinear one-dimensional time-fractional convection-diffusion-reaction equation (TFCDRE) with a variable coefficient. This method integrates the Crank-Nicolson finite difference scheme with extended cubic B-spline (ExCBS) basis functions. The Caputo fractional derivative is applied for temporal discretization, while the ExCBS functions are utilized for spatial discretization. The stability of the method was discussed by the von Neumann method, which shows unconditional stability; and the convergence analysis secure an order of $O(h^2+\triangle t^{2-\alpha})$. We demonstrated the efficiency and simplicity of our method through three numerical experiments and verified its accuracy using the absolute error $(L_2)$ and maximum error $(L_\infty)$ norms in temporal and spatial dimensions. The results are also graphically represented and show substantial agreement with the approximated and exact solution. We confirmed the effectiveness and accuracy of our approach over the local discontinuous Galerkin finite element and the compact finite difference methods, respectively, from the literature.

Correction to: Implicit-Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection-Diffusion Problems

Correction to: Implicit-Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection-Diffusion Problems

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

An ultraweak-local discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

This paper proposes and analyzes a fully discrete scheme for nonlinear biharmonic Schrödinger equations. We first write the single equation into a system of problems with second-order spatial derivatives and then discretize the space variable with an ultraweak discontinuous Galerkin scheme and the time variable with the Crank–Nicolson method. The proposed scheme proves to be computationally more efficient compared to the local discontinuous Galerkin method in terms of the number of equations needed to be solved at each single time step, and it is unconditionally stable without imposing any penalty terms. It also achieves optimal error convergence in L2 norm both in the solution and in the auxiliary variable with general nonlinear terms. We also prove several physically relevant properties of the discrete schemes, such as the conservation of mass and the Hamiltonian for the nonlinear biharmonic Schrödinger equations. Several numerical studies demonstrate and support our theoretical results.

Local Discontinuous Galerkin Methods with Multistep Implicit–Explicit Time Discretization for Nonlinear Schrödinger Equations

Local Discontinuous Galerkin Methods with Multistep Implicit–Explicit Time Discretization for Nonlinear Schrödinger Equations

Computational study based on the Laplace transform and local discontinuous Galerkin methods for solving fourth-order time-fractional partial integro-differential equations with weakly singular kernels

Computational study based on the Laplace transform and local discontinuous Galerkin methods for solving fourth-order time-fractional partial integro-differential equations with weakly singular kernels

Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of convection-diffusion type

We develop a local discontinuous Galerkin (LDG) method for a fourth-order singularly perturbed problem of convection-diffusion type. The existence and uniqueness of the computed solution are verified. Using the Shishkin mesh we derive an optimal-order energy-norm error estimate which is uniformly valid in the perturbation parameter. Numerical experiments are also given to support our theoretical findings.

Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system with Dougherty-Fokker-Planck collision operator

This paper develops energy-preserving discontinuous Galerkin (DG) methods for the Vlasov-Ampère (VA) system coupled with the Dougherty-Fokker-Planck (DFP) collision operator. While the classical VA system has been extensively studied, the inclusion of the collision operator introduces new challenges in conserving the total energy of the system. To address this, we design two energy-conserving temporal discretization methods: a second-order explicit scheme and a second-order implicit-explicit (IMEX) scheme. These schemes are coupled with the DG method, specifically using the local DG (LDG) method for the DFP part. We prove that the fully discrete schemes conserve the total particle number and total energy of the VA-DFP system at the fully discrete level. We further establish the L2 stability of the fully discrete explicit scheme. Numerical experiments are conducted to assess the accuracy, conservation property, and performance of the proposed schemes.

Note on quasi-optimal error estimates for the pressure for shear-thickening fluids

In this paper, we derive quasi-optimal a priori error estimates for the kinematic pressure for a Local Discontinuous Galerkin (LDG) approximation of steady systems of $p$-Navier--Stokes type in the case of shear-thickening, \textit{i.e.}, $p>2$, imposing a new Muckenhoupt regularity condition on the viscosity of the fluid, which is mild in the two-dimensional case but potentially restrictive in the three-dimensional case.

A structure-preserving local discontinuous Galerkin method for the stochastic KdV equation

This paper proposes a local discontinuous Galerkin (LDG) method for the stochastic Korteweg-de Vries (KdV) equation with multi-dimensional multiplicative noise. In the mean square sense, we show that the numerical method is L2 stable and it preserves energy conservation and energy dissipation. If the degree of the polynomial is n, the optimal error estimate in the mean square sense can reach as n+1. Finally, structure-preserving and convergence are verified by numerical experiments.

Local Discontinuous Galerkin Methods with Decoupled Implicit-Explicit Time Marching for the Growth-Mediated Autochemotactic Pattern Formation Model

Local Discontinuous Galerkin Methods with Decoupled Implicit-Explicit Time Marching for the Growth-Mediated Autochemotactic Pattern Formation Model

Error analysis for local discontinuous Galerkin semidiscretization of Richards’ equation

Abstract This paper concerns an error analysis of the space semidiscrete scheme for the Richards’ equation modeling flows in variably saturated porous media. This nonlinear parabolic partial differential equation can degenerate; namely, we consider the case where the time derivative term can vanish, i.e., the fast-diffusion type of degeneracy. We discretize the Richards’ equation by the local discontinuous Galerkin method, which provides high order accuracy and preserves stability. Due to the nonlinearity of the problem, special techniques for numerical analysis of the scheme are required. In particular, we combine two partial error bounds using continuous mathematical induction and derive a priori error estimates with respect to the spatial discretization parameter and the Hölder coefficient of the nonlinear temporal derivative. Finally, the theoretical results are supported by numerical experiments, including cases beyond the assumptions of the theoretical results.

High-Order Bound-Preserving Local Discontinuous Galerkin Methods for Incompressible and Immiscible Two-Phase Flows in Porous Media

High-Order Bound-Preserving Local Discontinuous Galerkin Methods for Incompressible and Immiscible Two-Phase Flows in Porous Media

An Ultra-weak Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for the KdV–Burgers–Kuramoto Equation

An Ultra-weak Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for the KdV–Burgers–Kuramoto Equation

Numerical methods for reinterpreted discrete fracture models with random inputs

This paper investigates the impact of uncertainty on the reinterpreted discrete fracture model (RDFM) for flow in porous media featuring fractures and barriers. The stochastic RDMF is formulated by parameterizing the uncertainty in the permeability, as well as the width of the fracture and barrier networks as random variables. To discretize these random variable, we adopt a stochastic collocation method based on a special choice of nodes. This collocation scheme offers optimal accuracy and easy implementation. The resulting collocation scheme is then combined with the local discontinuous Galerkin (LDG) method to accommodate the hybrid-dimensional Darcy’s law and the nature of the pressure/flux discontinuity. Additionally, a slope limiter is utilized to enhance stability. Numerical experiments are carried out to illustrate the accuracy and stability of the proposed scheme.

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.