Direct Discontinuous Galerkin Research Articles

A Direct Discontinuous Galerkin Method for Time Fractional Diffusion Equations with Fractional Dynamic Boundary Conditions

A Direct Discontinuous Galerkin Method for Time Fractional Diffusion Equations with Fractional Dynamic Boundary Conditions

Mixed Direct Discontinuous Galerkin Method for the Biharmonic Equation

In this paper, we use the mixed direct discontinuous Galerkin method (DDG) to solve the biharmonic equation. Firstly, by introducing an auxiliary variable, the biharmonic equation is split into two second-order equations. Secondly, the variational problem based on the DDG method of the system is derived and its well-posedness is proven. Next, error estimates of the approximate solution in L 2 norm and energy norm are present. For a given polynomial degree k (k ≥ 1), the optimal convergence rates concerning energy norm and norm are k and k + 1, respectively. Finally, numerical results demonstrate the accuracy and capability of the method.

Hp-Version direct discontinuous Galerkin method for elliptic problems

hp-Version direct discontinuous Galerkin method for elliptic problems

A direct discontinuous Galerkin method for a high order nonlocal conservation law

A direct discontinuous Galerkin method for a high order nonlocal conservation law

A Generalized Framework for Direct Discontinuous Galerkin Methods for Nonlinear Diffusion Equations

A Generalized Framework for Direct Discontinuous Galerkin Methods for Nonlinear Diffusion Equations

Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach

Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach

Gradient recovery based a posteriori error estimator for the adaptive direct discontinuous Galerkin method

Gradient recovery based a posteriori error estimator for the adaptive direct discontinuous Galerkin method

Coupling of direct discontinuous Galerkin method and natural boundary element method for exterior interface problems with curved elements

Coupling of direct discontinuous Galerkin method and natural boundary element method for exterior interface problems with curved elements

HODG:high-order discontinuous Galerkin methods for solving compressible Euler and Navier-Stokes equations - an open-source component-based development framework

In this work, we present HODG, an open-source component-based development framework based on high order Discontinuous Galerkin (DG) methods for solving compressible Euler and Navier-Stokes equations. This framework is written in pure C++11, and proposes “component” as the basic function unit, which is the key to the Interface-Oriented Programming principle and Aspect-Oriented Programming (AOP) technology. Built on the top-level design of components, HODG is a flexible yet pragmatic development framework that works right out of the box and is easy to use for starters and developers.The current release of HODG supports structured, unstructured, hybrid, and second-order meshes available. It is capable of solving Euler and Navier-Stokes (N-S) equations in two dimensions and three dimensions. In this development framework, spatial accuracy is easy to be extended to higher orders, and DG-P0, DG-P1, DG-P2 (up to third order) are implemented in the current version. Moreover, implicit temporal discretization and explicit discretization are available. Especially for viscous fluxes, the direct Discontinuous Galerkin (DDG) formulation and Bassi and Rebay II (BR2) scheme are implemented. In the case of strong discontinuities or shock waves, artificial viscosity is applied to capture shock waves. Various benchmark numerical examples are provided to demonstrate the full capabilities of HODG. The software is freely available under an MIT license. Program summaryProgram Title: HODG v2.0.0CPC Library link to program files:https://doi.org/10.17632/835732yz8s.1Code Ocean capsule:https://codeocean.com/capsule/4788938Licensing provisions: MITProgramming language: C++11Nature of problem: Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.Solution method: DG-P0, DG-P1, DG-P2 spatial discretization, combining with DDG and BR2 schemes for viscous fluxes, explicit and implicit time schemes, and supporting structured, unstructured, hybrid, and high-order meshes.unusual features: developed based on modern C++11 features; propose the “component” as basic function unit, which follows the practice of Interface-Oriented Programming principle; Special component named “action” is used, and is the key to AOP technology in this framework; Implicitly implements the Dependency Injection (DI) technology; Flexible yet pragmatic, and works right out of the box.

Direct discontinuous Galerkin method for potential magnetic field solutions

In this paper, we employ the direct discontinuous Galerkin (DDG) method for the first time to extrapolate the coronal potential magnetic field (PF) with the source surface (SS) and call the developed numerical model as the DDG-PFSS solver. In this solver, the Laplace’s equation is solved by means of the time-dependent method, i.e., introducing a pseudo-time term into the Laplace’s equation and changing the boundary value problem into the initial-boundary value problem. The steady-state solution of the initial-boundary value problem is the solution of the Laplace’s equation to be solved. This formulation facilitates the implementation of the DDG discretization. In order to validate the DDG-PFSS solver, we test a problem with the exact solution, which demonstrates the effectiveness and third-order accuracy of the solver. Then we apply it to the extrapolation for the coronal potential magnetic field. We use the integral GONG synoptic magnetogram of Carrington rotation (CR) 2060 as the boundary condition and achieve the global potential magnetic field solution by the DDG-PFSS solver. The numerical results such as the coronal holes and streamer belts derived from the DDG-PFSS solver are in good agreement with those obtained from the spherical harmonic expansion method. Also, based on the numerical magnetic field and Wang-Sheeley-Arge model, the obtained solar wind speed is found to basically capture the structures of the high- and low-speed streams observed at 1 AU. These results suggest that the DDG-PFSS solver can be seen as a contribution to the numerical methods for obtaining the global potential magnetic field solutions of the solar corona.

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined as a function of the exact solution and approximated using the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables' gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P1, P2, P3 and P4 polynomial approximations are performed to verify the optimal (k+1)th high-order accuracy of the method. The new DDGIC method is shown to be able to accurately calculate physical quantities such as lift, drag, and friction coefficients as well as separation angle and Strouhal number.

Adaptive direct discontinuous Galerkin method for elliptic equations

In this paper, adaptive direct discontinuous Galerkin method for second order elliptic equations is studied in two dimensional. A numerical flux with general weighted averages and proper weights for interface problems is introduced. In addition to the optimal a priori error estimator, we propose a residual-type a posteriori error estimator. The global upper bound and local lower bound for the error in DG norm are established. Several numerical examples are carried out to confirm the reliability and efficiency of the proposed error estimator and the corresponding adaptive direct discontinuous Galerkin method.

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Diffusion Problems

In this paper, a fully discrete stability analysis is carried out for the direct discontinuous Galerkin (DDG) methods coupled with Runge-Kutta-type implicit-explicit time marching, for solving one-dimensional linear convection-diffusion problems. In the spatial discretization, both the original DDG methods and the refined DDG methods with interface corrections are considered. In the time discretization, the convection term is treated explicitly and the diffusion term implicitly. By the energy method, we show that the corresponding fully discrete schemes are unconditionally stable, in the sense that the time-step $$\tau$$ is only required to be upper bounded by a constant which is independent of the mesh size h. Optimal error estimate is also obtained by the aid of a special global projection. Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

Analysis of direct discontinuous Galerkin methods for multi-dimensional convection–diffusion equations

We provide a framework for the analysis of the direct discontinuous Galerkin (DDG) methods for multi-dimensional convection–diffusion equations subject to various boundary conditions. A key tool is the global projection constructed by the DDG scheme applied to an associated elliptic problem. Such projection is well-defined for a class of diffusive flux parameters, and the optimal projection error in $$L^2$$ is obtained with an arbitrary locally regular partition of the domain and for an arbitrary degree of polynomials. This results in the optimal $$L^2$$ error for the DDG method to the elliptic problem, and further leading to the optimal $$L^2$$ error for the DDG method to the convection–diffusion problem.

Third order positivity-preserving direct discontinuous Galerkin method with interface correction for chemotaxis Keller-Segel equations

In this article, we apply direct discontinuous Galerkin method with interface correction (DDGIC) to solve chemotaxis Keller-Segel equation and prove the quadratic polynomial solution satisfying positivity-preserving with third order of accuracy. We show DDGIC method can obtain optimal convergence order for the cell density variable without introducing extra auxiliary variables to approximate the gradient of the chemical concentration. This is due to the super convergent property of the DDGIC method. We prove that, with the proper choice of the numerical flux coefficients, the cell density approximation can be preserved none negative at all time. Uniform third order of accuracy is maintained with the positivity-preserving limiter applied. For the chemotaxis model with blow up or singular solutions, the DDGIC method can effectively remove negative values approximations and accurately capture the blow up time.

A reconstructed discontinuous Galerkin method for incompressible flows on arbitrary grids

The discontinuous Galerkin (DG) methods have attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. However, the DG methods have their own weakness due to the high computational costs and storage requirements. In this work, we develop a high-order hybrid reconstructed DG (rDG) method for solving the INS equations on arbitrary grids. To be specific, the inviscid term of the INS equations is discretized by applying the third-order hybrid rDG(P1P2) method with a simplified artificial compressibility flux, while the viscous term of the INS equations is discretized by using the simple direct DG (DDG) method. A number of incompressible flow problems, in both steady and unsteady forms, for a variety flow conditions are computed to numerically assess the performance of the hybrid rDG(P1P2) method, which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs. Furthermore, a detailed comparison of a variety of different reconstructed strategies is performed and presented. Numerical results demonstrate that the hybrid rDG(P1P2) method outperforms the rDG(P1P2) method based on either the original least-squares reconstruction or the Green-Gauss reconstruction for solving the INS equations.

Exponential boundary-layer approximation space for solving the compressible laminar Navier-Stokes equations

In general, it needs to take about nearly 10 grid points inside a wall boundary layer for low accuracy order methods to get satisfactory wall-normal gradient related results, such as friction coefficient and wall heat flux. If there exist extreme points inside the boundary layer, this situation becomes even worse. In this work, with the help of the analytic solution of the one-dimensional steady compressible Navier-Stokes equations under some restrictions, we show that the flow variables actually vary in the form of an exponential function instead of a polynomial one inside the boundary layer. Then, we propose an exponential space for approximating the solutions inside the boundary layer and numerically implementing it in the frame of direct discontinuous Galerkin (DDG) method. We show that the DDG methods based on the exponential boundary-layer space give much better numerical results for both conservative variables and wall-normal gradients than those with the standard polynomial space. Generally only 1–2 grid points inside the boundary layer are demanded to resolve the wall boundary layer to obtain satisfactory wall-normal gradients under the exponential space. Preliminary extension to two-dimensional laminar boundary-layer flow shows a similar performance of the proposed exponential boundary-layer space, exhibiting its potential applications in high dimensions.

High order symmetric direct discontinuous Galerkin method for elliptic interface problems with fitted mesh

In this article, we aim to develop a high order symmetric direct discontinuous Galerkin method solving elliptic interface problems with zero or non-zero solution jump and flux jump interface conditions. We focus on the case, in which the mesh is partitioned along the curved interface. The two interface jump conditions are naturally and simultaneously built into the numerical flux definition on the curved triangular elements' edges that overlap with the interface. We obtain a stable and high order method, regardless of the combination of the two interface jump conditions. The two interface jump conditions are both weakly enforced in the scheme formulation. Optimal (k+1)th order L2 norm error estimate is proved for polygonal interfaces. A sequence of numerical examples is carried out to verify the optimal convergence of the symmetric direct discontinuous Galerkin method with high order P2, P3 and P4 approximations. Uniform optimal convergence order, which is independent of the diffusion coefficient ratio inside and outside of the interface, is obtained. The symmetric direct discontinuous Galerkin method is shown to be capable of handling interface problems with complicated geometries.

The Error Estimates of Direct Discontinuous Galerkin Methods Based on Upwind-Baised Fluxes

In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.