Local Discontinuous Galerkin Method Research Articles

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.

A hybrid finite difference level set–implicit mesh discontinuous Galerkin method for multi-layer coating flows

A mathematical model and numerical framework are presented for computing multi-physics multi-layer coating flow dynamics, with applications to the leveling of multi-layer paint films. The algorithm combines finite difference level set methods and high-order accurate sharp-interface implicit mesh discontinuous Galerkin methods to capture a complex set of multi-physics, incorporating Marangoni-driven multi-phase interfacial flow and the transport, mixing, and evaporation of multiple dissolved species. In particular, we develop several numerical methods for this multi-physics problem, including: high-order local discontinuous Galerkin methods for Poisson problems with Robin boundary conditions on implicitly-defined domains, to capture solvent evaporation; finite difference surface gradient methods, to robustly and accurately incorporate Marangoni stresses; and a coupled multi-physics time stepping approach, to incorporate all the different solvers at play including quasi-Newtonian fluid flow. The framework is applicable to an arbitrary number of layers and dissolved species; here, we apply it in a variety of settings, including multi-solvent evaporative paint dynamics, the flow and leveling of multi-layer automobile paint coatings in both 2D and 3D, and an examination of interfacial turbulence within a multi-layer matter cascade. Our results reproduce several phenomena observed in experiment, such as the formation of Marangoni plumes and Bénard cells. We also use the model to study the impact of long-wave deformational surface modes on immersed interfaces as well as the emergence of the final multi-layer film profile.

Local discontinuous Galerkin method coupled with the implicit‐explicit Runge–Kutta method for the time‐dependent micropolar fluid equations

AbstractIn this article, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal implicit‐explicit Runge–Kutta (RK) evolution for the micropolar fluid equations are adopted to construct the discretization method. To avoid the incompressibility constraint, the artificial compressibility strategy method is used to convert the micropolar fluid equations into the Cauchy–Kovalevskaja type equations. Then the LDG method based on the modal expansion and the implicit‐explicit RK method are properly combined to construct the expected third‐order method. Theoretically, the unconditionally stable of the fully discrete method are derived in multidimensions for triangular meshs. And the numerical experiments are given to verify the theoretical and effectiveness of the presented methods.

A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

Supercloseness of the LDG method for a singularly perturbed convection–diffusion problem on Bakhvalov-type mesh

In this study, the focus is on exploring the supercloseness property of the local discontinuous Galerkin (LDG) method in the context of a singularly perturbed convection–diffusion problem on Bakhvalov-type mesh. By developing specialized local Gauss–Radau projections in the two-dimensional case, and establishing a novel interpolation premised on the special projections, supercloseness of an optimal order k+1 can be achieved on Bakhvalov-type mesh. Notably, the obtained result remains unaffected by the singular perturbation parameter ɛ, indicating that the limitations encountered in previous studies, where the supercloseness results on Bakhvalov-type mesh were always impacted by the logarithmic factor ln1ɛ are overcome.

A discontinuous Galerkin finite element model for compound flood simulations

Many tropical cyclones, e.g., Hurricane Harvey (2017), have lead to significant rainfall and resulting runoff. When the runoff interacts with storm surge, the resulting floods can be greatly amplified and lead to effects that cannot be correctly modeled by simple superposition of its distinctive sources. In an effort to develop accurate numerical simulations of runoff, surge, and compounding floods, we develop a local discontinuous Galerkin method for modified shallow water equations. In this modification, nonzero sources to the continuity equation are included to incorporate rainfall into the model using parametric rainfall models from literature as well as hindcast data. The discontinuous Galerkin spatial discretization is accompanied with a strong stability preserving explicit Runge Kutta time integrator. Hence, temporal stability is ensured through the Courant–Friedrichs–Lewy (CFL) condition and we exploit the embarrassingly parallel nature of the developed method using MPI parallelization.We demonstrate the capabilities of the developed method though a sequence of physically relevant numerical tests, including small scale test cases based on laboratory measurements and large scale experiments with Hurricane Harvey in the Gulf of Mexico. The results highlight the conservation properties and robustness of the developed method and show the potential of compound flood modeling using our approach.

Application of the Local Discontinuous Galerkin Method to the Solution of the Quasi-Gas Dynamic System of Equations

Application of the Local Discontinuous Galerkin Method to the Solution of the Quasi-Gas Dynamic System of Equations

Optimal Error Estimates of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Kdv Type Equations

Optimal Error Estimates of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Kdv Type Equations

Stability analysis and error estimates of implicit Runge-Kutta local discontinuous Galerkin methods for linear bi-harmonic equation

Stability analysis and error estimates of implicit Runge-Kutta local discontinuous Galerkin methods for linear bi-harmonic equation

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

A fourth-order singularly perturbed problem of reaction–diffusion type is solved numerically by a local discontinuous Galerkin (LDG) method. Under suitable hypotheses, we prove optimal convergence of the LDG method on a Shishkin mesh; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, we obtain O((N−1lnN)k+1/2) convergence in the energy norm. The error bound is uniformly valid with respect to the singular perturbation parameter. In the error analysis, we exploit a relationship between the numerical solution of the third-order derivative with the gradient, the numerical solution and its element interface jump. We discuss also the convergence of the LDG method on two Bakhvalov-type meshes. Numerical experiments indicate that our error estimate is sharp.

A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations

AbstractThis paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.

An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is studied for a one-dimensional singularly perturbed reaction-diffusion problem. Based on the discrete Green's function, we derive some improved pointwise error estimates in the regular and layer regions. For the LDG approximation to the solution itself, the convergence rate of the pointwise error in the regular region is sharp. For the LDG approximation to the derivative of the solution, the convergence rate of the pointwise error in the whole domain is optimal. We establish also optimal pointwise error estimates in the case that the regular component of the exact solution belongs to the finite element space. Numerical experiments are presented to validate the theoretical results.

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.

L1/LDG algorithm for time‐Caputo space‐Riesz fractional convection equation in two dimensions

This paper proposes an L1/LDG algorithm for a time–space fractional convection equation on a bounded rectangular domain . Here, the temporal partial derivative is in the sense of Caputo with order , and the spatial partial derivatives are of Riesz type with orders in ‐ and ‐directions, respectively. L1 approximation is utilized to evaluate the temporal Caputo derivative, while the local discontinuous Galerkin (LDG) method is adopted to deal with the spatial Riesz derivative. The rigorous numerical stability analysis and error estimate are presented which are supported by illustrative numerical examples.

Stability analysis and error estimates of local discontinuous Galerkin method for nonlinear fractional Ginzburg–Landau equation with the fractional Laplacian

Stability analysis and error estimates of local discontinuous Galerkin method for nonlinear fractional Ginzburg–Landau equation with the fractional Laplacian

A three-dimensional high-order flux reconstruction lattice boltzmann flux solver for incompressible laminar and turbulent flows

A three-dimensional (3D) high-order flux reconstruction lattice Boltzmann flux solver (FR-LBFS) is developed in this paper for accurate and efficient simulations of incompressible laminar and turbulent flows. The original lattice Boltzmann flux solver (LBFS) is a second-order finite volume method (FVM) which combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. But it is not convenient to construct high-order FVM, and the compactness is always a bottleneck. The present work separates LBFS from FVM and combines LBFS with high-order FR scheme. Thus, the FR-LBFS inherits the superiority of LBFS and FR and shares the following attractive features: (i) a fully-explicit arbitrary order compact method, (ii) weakly compressible (WC) model for unsteady incompressible flows and (iii) evaluating inviscid and viscous fluxes simultaneously and no particular technique requirement, such as local discontinuous Galerkin method (LDG), for viscous term discretization. To validate the current method, various 3D incompressible laminar isothermal and thermal numerical experiments are presented first. Good agreements are achieved between the current results and the previous experimental and numerical data whilst the present high-order solver adopt coarse meshes. Furthermore, the ability of the proposed method to perform accurate and stable computations of incompressible decaying homogeneous isotropic turbulent flows and turbulent channel flows with different mesh resolutions and accuracy orders are investigated. The results show that the present 3D FR-LBFS is a promising tool for under-resolved or implicit large eddy simulation (ILES) of incompressible turbulent flows.