Discontinuous Galerkin Method In Space Research Articles

Comparison of implicit time-discretization schemes for hybridized discontinuous Galerkin methods

The present study is focused on the application of two families of implicit time-integration schemes for general time-dependent balance laws of convection-diffusion-reaction type discretized by a hybridized discontinuous Galerkin method in space, namely backward differentiation formulas (BDF) and diagonally implicit Runge-Kutta (DIRK) methods. Special attention is devoted to embedded DIRK methods, which allow the incorporation of time step size adaptation algorithms in order to keep the computational effort as low as possible. The properties of the numerical solution, such as its order of convergence, are investigated by means of suitably chosen test cases for a linear convection-diffusion-reaction equation and the nonlinear system of Navier-Stokes equations. For problems considered in this work, the DIRK methods prove to be superior to high-order BDF methods in terms of both stability and accuracy.

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>

Arbitrary order DG-DGLM method for hyperbolic systems of multi-dimensional conservation laws

An arbitrary order discontinuous Galerkin method in space and time is proposed to approximate the solution to hyperbolic systems of multi-dimensional conservation laws. Weak formulation is derived through the definition of weak divergence. Weak solution is given as a pair of weak functions on the element and the edge, respectively. Weak solution on the edge is characterized as the average of the solutions on the elements sharing the edge. Stability of the approximate solution is proved in a broken L2(L2) norm and also in a broken l∞(L2) norm. Error estimates of O(hr+knq) with Pr(E) and Pq(Jn) elements (r,q>1+d2) are then derived in a broken L2(L2) norm, where h and kn are the maximum diameters of the elements and the time step of Jn, respectively, Jn is the time interval, and d is the dimension of the spatial domain.

A high‐order implicit hybridizable discontinuous Galerkin method for the Benjamin‐Bona‐Mahony equation

SummaryThis article introduces an efficient numerical scheme for the solution of the Benjamin‐Bona‐Mahony equation. The scheme is fully implicit, and combines high‐order time stepping and a hybridizable discontinuous Galerkin method in space. Implicit time stepping allows for larger time steps, while remaining robust for long‐time simulations. It is shown that the scheme is energy stable, and conserves mass. Computational experiments demonstrate that the scheme is high‐order accurate, robust, and possesses good conservation properties.

Systematic development of upwind numerical fluxes for the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media with physical interfaces

This research work presents, within a unified and wave oriented variational framework, systematic development of upwind numerical fluxes for the space discontinuous Galerkin methods to model elastic wave propagation in multidimensional anisotropic media with discontinuous material properties. Both first-order velocity–stress and velocity–strain wave formulations are considered. The proposed approach allows the derivation of upwind numerical fluxes in a well structured and hierarchical way according to the degree of inhomogeneity across a physical interface. The numerical fluxes that are exact solutions of a relevant Riemann problem defined at a physical interface are obtained. The developed explicit and intrinsic tensorial expressions of upwind numerical fluxes in multidimensional case allow a better understanding and analysis of the physical meaning of involved terms. As numerical applications, an example with a physical interface separating two materials, one anisotropic and the other isotropic, and an example of polycrystalline material that presents a particular case with a larger number of physical interfaces, are considered. The proposed numerical fluxes are numerically investigated and validated.

Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation

In this paper, an accurate numerical method is presented to solve a class of variable-order fractional diffusion problem. The problem first is discretized by a finite difference method in temporal direction, and then a local discontinuous Galerkin method in space. The stability and L2 convergence of the proposed scheme are derived for all variable-order α(t)∈(0,1). We prove that the scheme is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Some numerical experiments are provided to verify the theoretical analysis and high-accuracy of the proposed method.

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.

Analysis of local discontinuous Galerkin method for time-space fractional sine-Gordon equations

The aim of the present paper is to introduce a numerical method for time-space fractional sine-Gordon equation. The fractional derivative on the space and on the time are considered in the sense of Reimann–Liouville (of order 1≤β≤2) and in the sense of Caputo (of variable order 1≤α(t)<2), respectively. The basic idea is to apply local discontinuous Galerkin method in space and a finite difference method in time. The stability and convergence analysis of the method are presented. Numerical results show that the accuracy and reliability of the proposed method for time-space fractional sine-Gordon equation.

An arbitrary high order series-based time-marching method with discontinuous Galerkin method for 2D convection problem

This paper presents an arbitrary high-order series-based time-marching (SBTM) method to match with the high-order discontinuous Galerkin (DG) method in spatial discretization. And the detail of this hybrid method, SBTM-DG, is presented by solving a 2D convection problem where the exact solution of the semi-discrete system derived by DG method can be obtained by SBTM method. The SBTM-DG method is explicit single-step scheme of arbitrary high order of accuracy in time and with all advantages of DG method in space. The high order of accuracy in temporal direction is obtained by multiplications between small-scale matrixes and vectors in each cells with dimension K=(k+1)(k+2)/2, where k is the degree of basic function of DG method, which makes the SBTM-DG method memory-saving and efficient. With the help of projection, the error estimation between the exact solution and the full-discrete solution is O(h)k+1+O(Δt)S, where S is the degree of polynomial of time variable. By energy method we can prove the stability. By matrix method, we obtain the stability condition where time step Δt can be larger than mesh size h. Numerical examples confirm the analysis and show the good performance of SBTM-DG method. Furthermore, the convergence orders are around 20 about k, which makes SBTM-DG method significant.

Some comparisons and analyses of time or space discontinuous Galerkin methods applied to elastic wave propagation in anisotropic and heterogeneous media

This research work presents some comparisons and analyses of the time discontinuous space–time Galerkin method and the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media. Mechanism of both methods to ensure their stability using time or space discontinuities of unknown fields is analyzed and compared. The most general case of anisotropic and heterogeneous media with physical interfaces of discontinuous material properties is considered, especially for the space discontinuous Galerkin method. A new stability result is proved. Numerical applications to different elastic media, more particularly polycrystalline materials containing a large number of physical interfaces, are also presented to confirm theoretical analyses.

A parabolic level set reinitialisation method using a discontinuous Galerkin discretisation

Level set reinitialisation is a part of the level set methodology which allows one to generate, at any point during level set evolution, a level set function which is a signed distance function to its own zero isocontour. Whilst not in general a required condition, maintaining the level set function as a signed distance function is often desirable as it removes a known source of numerical instability. This paper presents a novel level set reinitialisation method based on the solution of a nonlinear parabolic PDE. The PDE is discretised using a symmetric interior penalty discontinuous Galerkin method in space, and an implicit Euler method in time. Also explored are explicit and semi-implicit time discretisations, however, numerical experiments demonstrate that such methods suffer from severe time step restrictions, leading to prohibitively large numbers of iterations required to achieve convergence. The proposed method is shown to be high-order accurate through a number of numerical examples. More specifically, the presented experimental orders of convergence align with the well established optimal convergence rates for the symmetric interior penalty method; that is the error in the L2 norm decreases proportionally to hp+1 and the error in the DG norm decreases proportionally to hp.

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

Comparison of numerical methods for the solution of viscous incompressible and low Mach number compressible flow

The novelty of the paper is the application of the discontinuous Galerkin method (DGM) to the solution of viscous compressible flow with low Mach number (M≈10−2−10−3). The compressible Navier–Stokes equations are written in the conservative form using conservative variables. They are discretized by the DGM in space. In time we use either the backward difference formula (BDF) or the discontinuous Galerkin method. The resulting technique is called the space-time discontinuous Galerkin method (STDGM).The goal of the presented analysis is to prove the ability of the developed DGM to solve viscous compressible low Mach number flows described by the compressible Navier–Stokes equations without any modification and to obtain results comparable with simulations of incompressible viscous flow described by the incompressible Navier–Stokes equations, solved by the finite element method (FEM). We present numerical results for a flow in a simplified channel of the human vocal tract, for a flow around the airfoil NACA0012 and moreover for flow-induced airfoil vibrations. The results prove the robustness of the discontinuous Galerkin method with respect to the low Mach number.

Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order

In this paper, a numerical method is proposed for solving distributed order diffusion equation, which arises in the mathematical modeling of ultra-slow diffusion processes observed in some physical problems, whose solution decays logarithmically as the time t tends to infinity. Based on local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with the order $$O(h^{k+1}+\Delta t+\Delta \alpha ^2)$$ , where $$h, \Delta t$$ , $$\Delta \alpha $$ and k are the step size in space, time, distributed order and the degree of piecewise polynomials, respectively. Extensive numerical examples are carried out to illustrate the effectiveness of the numerical schemes.

The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media

In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for the two-and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a divide-and-conquer framework. Several numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solving heterogeneous media problems on coarse meshes.

A Fully Discrete LDG Method for the Distributed-Order Time-Fractional Reaction–Diffusion Equation

In this paper, the numerical approximation of the distributed-order time-fractional reaction–diffusion equation is proposed and analyzed. Based on the finite difference method in time and local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with order $$O\left( h^{k+\frac{1}{2}}+(\Delta t)^2+\Delta \alpha ^4\right) $$ , where $$h,k,\Delta t$$ and $$\Delta \alpha $$ are the space-step size, piecewise polynomial degree, time-step size, step size in distributed-order variable, respectively. Numerical examples are presented to show the effectiveness and the accuracy of the numerical scheme.

Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method

This paper is devoted to determining a space-dependent source term in an inverse problem of the time-fractional diffusion equation. We use a method based on a finite difference scheme in time and a local discontinuous Galerkin method in space and investigate the numerical stability and convergence of the proposed method. Finally, various numerical examples are used illustrate the effectiveness and accuracy of the method.

A prioridiffusion-uniform error estimates for nonlinear singularly perturbed problems: BDF2, midpoint and time DG

This work deals with a nonlinear nonstationary semilinear singularly perturbed convection-diffusion problem. We discretize this problem by the discontinuous Galerkin method in space and by the midpoint rule, BDF2 and quadrature variant of discontinuous Galerkin in time. We present a priori error estimates for these three schemes that are uniform with respect to the diffusion coefficient going to zero and valid even in the purely convective case.

Strong convergence of the discontinuous Galerkin scheme for the low regularity miscible displacement equations

Strong convergence of the numerical solution to a weak solution is proved for a nonlinear coupled flow and transport problem arising in porous media. The method combines a mixed finite element method for the pressure and velocity with an interior penalty discontinuous Galerkin method in space for the concentration. Using functional tools specific to broken Sobolev spaces, the convergence of the broken gradient of the numerical concentration to the weak solution is obtained in the L2 norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 489–513, 2017

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

In this paper a new method for the discretization of the radiation transport equation is presented, based on a discontinuous Galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization. To cope with the discontinuous spatial nature of the solution, a generalized Riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes. A new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem. The resulting numerical fluxes for the various possible cases where neighboring elements have an equal, higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation. The choice of discontinuous trial functions not only has the advantage of easing local refinement, it also facilitates the use of efficient sweep-based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution. The approach is illustrated by a series of numerical experiments. Results show high orders of convergence for the scalar flux on angular refinement. The generalized Riemann upwinding procedure leads to stable and consistent solutions. Further the sweep-based solver performs well when used as a preconditioner for a Krylov method.