Discontinuous Galerkin Spectral Element Method Research Articles

High-Order Local Time Stepping on Moving DG Spectral Element Meshes

We derive and evaluate an explicit local time stepping (LTS) integration for the discontinuous Galerkin spectral element method on moving meshes. The LTS procedure is derived from Adams---Bashforth multirate time integration methods. We also present speedup and memory estimates, which show that the explicit LTS integration scales well with problem size. Time-step refinement studies with static and moving meshes show that the approximations are spectrally accurate in space and have design temporal accuracy. The numerical tests validate theoretical estimates that the LTS procedure can reduce computational cost by as much as an order of magnitude for time accurate problems.

A sparse and high-order accurate line-based discontinuous Galerkin method for unstructured meshes

We present a new line-based discontinuous Galerkin (DG) discretization scheme for first- and second-order systems of partial differential equations. The scheme is based on fully unstructured meshes of quadrilateral or hexahedral elements, and it is closely related to the standard nodal DG scheme as well as several of its variants such as the collocation-based DG spectral element method (DGSEM) or the spectral difference (SD) method. However, our motivation is to maximize the sparsity of the Jacobian matrices, since this directly translates into higher performance in particular for implicit solvers, while maintaining many of the good properties of the DG scheme. To achieve this, our scheme is based on applying one-dimensional DG solvers along each coordinate direction in a reference element. This reduces the number of connectivities drastically, since the scheme only connects each node to a line of nodes along each direction, as opposed to the standard DG method which connects all nodes inside the element and many nodes in the neighboring ones. The resulting scheme is similar to a collocation scheme, but it uses fully consistent integration along each 1-D coordinate direction which results in different properties for nonlinear problems and curved elements. Also, the scheme uses solution points along each element face, which further reduces the number of connections with the neighboring elements. Second-order terms are handled by an LDG-type approach, with an upwind/downwind flux function based on a switch function at each element face. We demonstrate the accuracy of the method and compare it to the standard nodal DG method for problems including Poisson’s equation, Euler’s equations of gas dynamics, and both the steady-state and the transient compressible Navier–Stokes equations. We also show how to integrate the Navier–Stokes equations using implicit schemes and Newton–Krylov solvers, without impairing the high sparsity of the matrices.

Explicit discontinuous Galerkin methods for unsteady problems

In this work we consider a special implementation of a discontinuous Galerkin (DG) method for general unstructured hexahedral element meshes called the discontinuous Galerkin Spectral Element Method (DGSEM). We are solving the compressible Navier–Stokes equations for unsteady turbulent flow simulations. We use explicit time stepping because of the high parallel scalability and also because the physical time scale of the simulation is in the range of the explicit time step restriction. In the explicit DGSEM framework, the efficiency of element-wise operations is highly improved compared to standard DG implementations. This improvement is due to collocated interpolation and integration points and tensor product nodal basis functions inside the hexahedron. In the first part of this paper, we describe the DGSEM scheme and derive the element-wise operators. We will conclude this part with accuracy and convergence analysis. The locality of the explicit DGSEM scheme is highly attractive for parallel computing, thus the second part is dedicated to a parallel performance analysis of the code. In the last part, we show the applicability of the scheme with a direct numerical simulation of a weak turbulent flow past a sphere at Reynolds number 1000.

Boundary states at reflective moving boundaries

We derive and evaluate boundary states for Maxwell’s equations, the linear, and the nonlinear Euler gas-dynamics equations to compute wave reflection from moving boundaries. In this study we use a Discontinuous Galerkin Spectral Element method (DGSEM) with Arbitrary Lagrangian–Eulerian (ALE) mapping for the spatial approximation, but the boundary states can be used with other methods, like finite volume schemes. We present four studies using Maxwell’s equations, one for the linear Euler equations, and one more for the nonlinear Euler equations. These are: reflection of light from a plane mirror moving at constant velocity, reflection of light from a moving cylinder, reflection of light from a vibrating mirror, reflection of sound from a plane wall and dipole sound generation by an oscillating cylinder in an inviscid flow. The studies show that the boundary states preserve spectral convergence in the solution and in derived quantities like divergence and vorticity.

Analysis of an $hp$-Nonconforming Discontinuous Galerkin Spectral Element Method for Wave Propagation

We analyze the consistency, stability, and convergence of an $hp$ discontinuous Galerkin spectral element method of Kopriva [J. Comput. Phys., 128 (1996), pp. 475--488] and Kopriva, Woodruff, and Hussaini [Internat. J. Numer. Methods Engrg., 53 (2002), pp. 105--122]. The analysis is carried out simultaneously for acoustic, elastic, coupled elastic-acoustic, and electromagnetic wave propagation. Our analytical results are developed for both conforming and nonconforming approximations on hexahedral meshes using either exact integration with Legendre--Gauss quadrature or inexact integration with Legendre--Gauss--Lobatto quadrature. A mortar-based nonconforming approximation is developed to treat both $h$- and $p$-nonconforming meshes simultaneously. The mortar approach is constructed in such a way that consistency, stability, and convergence analyses for nonconforming approximations follow the conforming counterparts with minimal modifications. In particular, it casts $hp$-nonconforming interfaces into equivalent conforming faces on mortars, for which the analyses are easily carried out using standard approaches. Sharp $hp$-convergence results are then proved for nonconforming approximations of time-dependent wave propagation problems using both exact and inexact quadratures.

A Comparison of the Dispersion and Dissipation Errors of Gauss and Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods

We examine the dispersion and dissipation properties of the Gauss and Gauss–Lobatto discontinuous Galerkin spectral element methods (DGSEMs), for linear wave propagation problems. We show that the inherent underintegration in the Gauss–Lobatto variant can be interpreted as a modal filtering of the highest polynomial mode. This in turn has a drastic impact on the dispersion and dissipation relations of the Gauss–Lobatto DGSEM compared to the Gauss variant. We show that the Gauss DGSEM is typically more accurate than the Gauss–Lobatto variant, needing fewer points per wavelength for a given accuracy while on the other hand being more restricted in the explicit time step choice. We show that the spectra of the DGSEM operators depend on the boundary conditions applied and that the ratio of the time step restrictions of the two schemes depends on the choice of boundary conditions.

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.

Discontinuous Galerkin spectral element approximations on moving meshes

We derive and evaluate high order space Arbitrary Lagrangian–Eulerian (ALE) methods to compute conservation laws on moving meshes to the same time order as on a static mesh. We use a Discontinuous Galerkin Spectral Element Method (DGSEM) in space, and one of a family of explicit time integrators such as Adams–Bashforth or low storage explicit Runge–Kutta. The approximations preserve the discrete metric identities and the Discrete Geometric Conservation Law (DGCL) by construction. We present time-step refinement studies with moving meshes to validate the approximations. The test problems include propagation of an electromagnetic gaussian plane wave, a cylindrical pressure wave propagating in a subsonic flow, and a vortex convecting in a uniform inviscid subsonic flow. Each problem is computed on a time-deforming mesh with three methods used to calculate the mesh velocities: from exact differentiation, from the integration of an acceleration equation, and from numerical differentiation of the mesh position.

A NON-SPURIOUS VECTOR SPECTRAL ELEMENT METHOD FOR MAXWELL'S EQUATIONS

In this paper, a non-spurious vector spectral element method is proposed to solve Maxwell's equations using E and H as variables. The mixed-order curl-conforming basis functions are used for both variables to facilitate applying boundary and interface conditions; and the interpolation degree of basis functions for E is set difierent from that for H to suppress the spurious modes. The proposed method can be utilized in both time domain and frequency domain, and it is very suitable for the future implementation of discontinuous Galerkin spectral element method. Numerical results demonstrate the property of spurious-free and the spectral accuracy of this method. The method has also been implemented for the more general flnite element method in time and frequency domains.

Numerical simulation of quench propagation at early phase by high-order methods

In this paper a high-order discontinuous Galerkin (DG) spectral element method (SEM) is introduced to deal with thermo-hydraulic quench simulation in superconducting magnets, specifically in the case of cable-in-conduit conductors (CICC). We will focus on 1D quench propagation at early phase which is important to understand the mechanism of quench propagation and protect the superconducting magnets. A second-order polynomial interpolation to the helium heat capacity is introduced in this study, and boundary conditions are imposed along the characteristics to overcome the typical difficulties encountered in numerical studies. Explicit fourth order Runge–Kutta time integration is used to match the high accuracy in space. Numerical results demonstrate the advantages in accuracy of this method when used to simulate early quench development.

Discontinuous Galerkin Spectral Element Simulation of a Thermal-Hydraulic Problem in Superconducting Magnets

ABSTRACT In this article we simulate one-dimensional quench propagation in superconducting magnets using cable-in-conduit conductors (CICCs) by a discontinuous Galerkin (DG) spectral element method (SEM) and explicit Runge-Kutta time integration. The supercritical helium flow in the CICC channel can be described by the Euler equations with an additional friction force and coupled heat transfer among the helium, the conductor, and the conduit. Roe's approximate Riemann solver for a real gas/fluid is used to compute numerical flux, and a nonreflecting boundary condition is introduced for the approximation of the outflow boundary condition. The method used here is highly parallelizable. Some numerical results are given and compared with those obtained by a finite-element method and with experimental data. This method shows high robustness to simulate the thermal-hydraulic problem with large nonlinear source terms.

Discontinuous Galerkin Spectral Element Simulation of Quench Propagation in Superconducting Magnets

In this paper we simulate 1D quench propagation in superconducting magnets using cable-in-conduit conductors (CICC) by a discontinuous Galerkin (DG) spectral element method (SEM) and explicit Runge-Kutta time integration. The supercritical helium flow is considered in the modeling of quench propagation in CICC, which can be expressed by the Euler equations with additional friction and coupled heat transfer between helium and conductor and conduit. Roe's approximate Riemann solver for real gas/fluid is used to compute numerical flux and nonreflecting boundary condition is introduced in the algorithm. The method used here is highly parallelizable. Some numerical results are given and compared with those obtained by another simulation method and experimental data.

Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element

AbstractDiscontinuous Galerkin spectral element method is used to solve the lattice Boltzmann equation (LBE) in the discrete velocity space. The triangular elements are adopted because of their flexibility to deal with complex geometries. The flow past a circular cylinder is simulated by the proposed scheme. The results are consistent with those obtained from the previous numerical methods and experiments. Copyright © 2003 John Wiley & Sons, Ltd.

Aircraft engine noise scattering by fuselage and wings: a computational approach

The paper presents a time-domain method for computation of sound radiation from aircraft engine sources to the far field. The effects of non-uniform flow around the aircraft and scattering of sound by fuselage and wings are accounted for in the formulation. The approach is based on the discretization of the inviscid flow equations through a collocation form of the discontinuous Galerkin spectral element method. An isoparametric representation of the underlying geometry is used in order to take full advantage of the spectral accuracy of the method. Large-scale computations are made possible by a parallel implementation based on message passing. Results obtained for radiation from an axisymmetric nacelle alone are compared with those obtained when the same nacelle is installed in a generic configuration, with and without a wing.