Discontinuous Galerkin Spatial Discretization Research Articles

A Variable Eddington Factor method for the 1-D grey radiative transfer equations with discontinuous Galerkin and mixed finite-element spatial differencing

The purpose of this paper is to present a Variable Eddington Factor (VEF) method for the 1-D grey radiative transfer equations that uses a lumped linear discontinuous Galerkin spatial discretization for the Snequations together with a constant-linear mixed finite-element discretization for the VEF moment and material temperature equations. The use of independent discretizations can be particularly useful for multiphysics applications such as radiation-hydrodynamics. The VEF method is quite old, but to our knowledge, this particular combination of differencing schemes has not been previously investigated for radiative transfer. We define the scheme and present computational results. As expected, the scheme exhibits second-order accuracy for the directionally-integrated intensity and material temperature, and behaves well in the thick diffusion limit. An important focus of this study is the treatment of the strong temperature dependence of the opacities and the spatial dependence of the opacities within each cell, which are not explicitly defined by the basic discretization schemes.

Filtered Discrete Ordinates Equations for Radiative Transport

We present and analyze a discrete ordinates ( $$\text {S}_N$$ ) discretization of a filtered radiative transport equation (RTE). Under certain conditions, $$\text {S}_N$$ discretizations of the standard RTE create numeric artifacts, known as “ray-effects”; the goal of the filter is to remove such artifacts. We analyze convergence of the filtered discrete ordinates solution to the solution of the non-filtered RTE, taking into account the effect of the filter as well as the usual quadrature and truncation errors that arise in discretize ordinate methods. We solve the filtered $$\text {S}_N$$ equations numerically with a discontinuous Galerkin spatial discretization and implicit time stepping. The form of the filter enables the resulting linear systems to be solved in an established Krylov framework. We demonstrate, via the simulation of two benchmark problems, the effectiveness of the filtering approach in reducing ray effects. In addition, we also examine efficiency of the method, in particular the balance between improved accuracy and additional cost of including the filter.

A parallel hp-adaptive high order discontinuous Galerkin method for the incompressible Navier-Stokes equations

We present a parallel hp-adaptive high order (spectral) discontinuous Galerkin method for approximation of the incompressible Navier-Stokes equations. The spatial discretization consists of equal-order polynomial approximations of the fluid velocity and pressure via discontinuous Galerkin spatial discretizations. For the nonlinear convective term we select the local Lax-Friedrichs flux, while for the divergence and gradient operators central fluxes are chosen. For the diffusive term, we use an interior penalty discontinuous Galerkin method to ensure stability and invertibility. The temporal discretization is an implicit-explicit Runge-Kutta method paired with a high-order splitting procedure to efficiently enforce the incompressibility condition at each time step. The compact stencil size, explicit time stepping of nonlinear terms, and inversion of sparse linear systems make the resulting method simple to parallelize while the local nature of the discontinuous Galerkin approximation makes hp-adaptive refinement natural to implement. We detail our implementation consisting of a tensor product basis of high order polynomials on quadrilateral elements, and implement hp-adaptivity using an inexpensive a posteriori error estimator to determine where refinement is necessary. p-Multigrid and pressure projection techniques are used to precondition the conjugate gradient linear solvers. We present several numerical tests to demonstrate the efficacy of the method, in particular in reducing the number of degrees of freedom needed and allocating computing resources to regions of sharp variation in transient incompressible Navier-Stokes flows.

High-order fully implicit solver for all-speed fluid dynamics

We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification of the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below $$M=10^{-5}$$ , which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the $${\mathrm{AUSM}}^{+}$$ -up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.

Global and local POD models for the prediction of compressible flows with DG methods

SummaryProper orthogonal decomposition (POD) allows to compress information by identifying the most energetic modes obtained from a database of snapshots. In this work, POD is used to predict the behavior of compressible flows by means of global and local approaches, which exploit some features of a discontinuous Galerkin spatial discretization. The presented global approach requires the definition of high‐order and low‐order POD bases, which are built from a database of high‐fidelity simulations. Predictions are obtained by performing a cheap low‐order simulation whose solution is projected on the low‐order basis. The projection coefficients are then used for the reconstruction with the high‐order basis. However, the nonlinear behavior related to the advection term of the governing equations makes the use of global POD bases quite problematic. For this reason, a second approach is presented in which an empirical POD basis is defined in each element of the mesh. This local approach is more intrusive with respect to the global approach but it is able to capture better the nonlinearities related to advection. The two approaches are tested and compared on the inviscid compressible flow around a gas‐turbine cascade and on the compressible turbulent flow around a wind turbine airfoil.

Stability Analysis and Error Estimates of Semi-implicit Spectral Deferred Correction Coupled with Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

In this paper, we focus on the theoretical analysis of the second and third order semi-implicit spectral deferred correction (SDC) time discretization with local discontinuous Galerkin (LDG) spatial discretization for the one-dimensional linear convection–diffusion equations. We mainly study the stability and error estimates of the corresponding fully discrete scheme. Based on the Picard integral equation, the SDC method is driven iteratively by either the explicit Euler method or the implicit Euler method. It is easy to implement for arbitrary order of accuracy. For the semi-implicit SDC scheme, the iteration and the left-most endpoint involved in the integral for the implicit part increase the difficulty of the theoretical analysis. To be more precise, the test functions are more complex and the energy equations are more difficult to construct, compared with the Runge–Kutta type semi-implicit schemes. Applying the energy techniques, we obtain both the second and third order semi-implicit SDC time discretization with LDG spatial discretization are stable provided the time step $$\tau \le \tau _{0}$$ , where the positive $$\tau _{0}$$ depends on the diffusion and convection coefficients and is independent of the mesh size h. We then obtain the optimal error estimates for the corresponding fully discrete scheme under the condition $$\tau \le \tau _{0}$$ with similar technique for stability analysis. Numerical examples are presented to illustrate our theoretical results.

An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form

We present an application of our general formulation (Appelö and Hagstrom (2015) [12]) to construct energy based, arbitrary order accurate, discontinuous Galerkin spatial discretizations of the linear elastic wave equation. The resulting methods are stable and, depending on the choice of numerical flux, conserve or dissipate the elastic energy. The performance of the method is demonstrated for problems with manufactured and exact solutions. Applications to more realistic problems are also presented. Implementations of the methods are freely available at Appelö and Hagstrom (2015) [19].

Third order implicit–explicit Runge–Kutta local discontinuous Galerkin methods with suitable boundary treatment for convection–diffusion problems with Dirichlet boundary conditions

Abstract To avoid the order reduction when third order implicit–explicit Runge–Kutta time discretization is used together with the local discontinuous Galerkin (LDG) spatial discretization, for solving convection–diffusion problems with time-dependent Dirichlet boundary conditions, we propose a strategy of boundary treatment at each intermediate stage in this paper. The proposed strategy can achieve optimal order of accuracy by numerical verification. Also by suitably setting numerical flux on the boundary in the LDG methods, and by establishing an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient and the given boundary conditions, we build up the unconditional stability of the corresponding scheme, in the sense that the time step is only required to be upper bounded by a suitable positive constant, which is independent of the mesh size.

Variable Eddington Factor Method for the SN Equations with Lumped Discontinuous Galerkin Spatial Discretization Coupled to a Drift-Diffusion Acceleration Equation with Mixed Finite-Element Discretization

ABSTRACTWe apply the Variable Eddington Factor (VEF) acceleration method to the 1-D, one-group SN equations with a Lumped Linear Discontinuous Galerkin (LLDG) spatial discretization in conjunction with a drift-diffusion acceleration equation discretized with the constant-linear Mixed Finite-Element Method (MFEM). While the VEF method is quite old, its application to this particular combination of SN and drift-diffusion discretizations is new. In addition to demonstrating the basic properties of our method, we investigate slope reconstruction techniques for improving the consistency between the SN and drift-diffusion solutions.

A priori and a posteriori analysis of the flow around a rectangular cylinder

The definition of a correct mesh resolution and modelling approach for the Large Eddy Simulation (LES) of the flow around a rectangular cylinder is recognized to be a rather elusive problem as shown by the large scatter of LES results present in the literature. In the present work, we aim at assessing this issue by performing an a priori analysis of Direct Numerical Simulation (DNS) data of the flow. This approach allows us to measure the ability of the LES field on reproducing the main flow features as a function of the resolution employed. Based on these results, we define a mesh resolution which maximize the opposite needs of reducing the computational costs and of adequately resolving the flow dynamics. The effectiveness of the resolution method proposed is then verified by means of an a posteriori analysis of actual LES data obtained by means of the implicit LES approach given by the numerical properties of the Discontinuous Galerkin spatial discretization technique. The present work represents a first step towards a best practice for LES of separating and reattaching flows.

Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

In this paper we study explicit peer methods with the strong stability preserving (SSP) property for the numerical solution of hyperbolic conservation laws in one space dimension. A system of ordinary differential equations is obtained by discontinuous Galerkin (DG) spatial discretizations, which are often used in the method of lines approach to solve hyperbolic differential equations. We present in this work the construction of explicit peer methods with stability regions that are designed for DG spatial discretizations and with large SSP coefficients. Methods of second- and third order with up to six stages are optimized with respect to both properties. The methods constructed are tested and compared with appropriate Runge–Kutta methods. The advantage of high stage order is verified numerically.

Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations

D. Seal acknowledges funding by the Naval Academy Research Council. The study of A. Jaust was supported by the Special Research Fund (BOF) of Hasselt University (Grant No. BOF16DOC02).

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction

In this work, we describe the implementation of an arbitrarily high-order hybrid solver for linear, kinetic, radiative transport equations. The hybrid method is derived from a splitting of the radiative flux into free-streaming and collisional components to which high- and low-resolution discrete ordinates methods are applied, respectively. Arbitrarily high orders of accuracy with respect to time and space are attained by combining an integral deferred correction (IDC) time integration scheme constructed with implicit Euler substepping on Radau II (right biased) nodes with an upwind discontinuous Galerkin (DG) spatial discretization on uniform Cartesian meshes. Numerical experiments are used to demonstrate that the aforementioned IDC methods can be constructed such that they are unconditionally stable (L-stable) to within machine precision. Asymptotic analysis is used to show that such IDC methods also preserve the diffusion limit of the underlying transport equation on the semi-discrete level, in the sense that the semi-discrete transport system under an implicit IDC discretization recovers the same IDC discretization of the limiting diffusion equation. Convergence results in one spatial dimension are provided, and it is found that while the hybrid method exhibits convergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approximation is comparable to a standard discrete ordinates approximation in many cases. Numerical results for two test problems in two spatial dimensions are given to compare the computational efficiency of the hybrid method against a standard discrete ordinates method, and to compare the efficiency of space–time discretizations of different orders of accuracy. The results indicate that a properly chosen hybrid discrete ordinates method can be more efficient than a standard discrete ordinates method by a factor of between 2 and 8 when IDC timestepping methods are used. Further, among the first-, third-, and fifth-order space–time discretizations considered here, the first-order method is the least efficient and the third-order method is the most efficient.

Simulating Magnetohydrodynamic Instabilities with Conservative Perturbed MHD Model Using Discontinuous Galerkin Method

AbstractIn magnetically confined plasma research, the understandings of small and large perturbations at equilibrium are both critical for plasma controlling and steady state operation. Numerical simulations using original MHD model can hardly give clear picture for small perturbations, while non-conservative perturbed MHD model may break conservation law, and give unphysical results when perturbations grow large after long-time computation. In this paper, we present a nonlinear conservative perturbed MHD model by splitting primary variables in original MHD equations into equilibrium part and perturbed part, and apply an approach in the framework of discontinuous Galerkin (DG) spatial discretization for numerical solutions. This enables high resolution of very small perturbations, and also gives satisfactory non-smooth solutions for large perturbations, which are both broadly concerned in magnetically confined plasma research. Numerical examples demonstrate satisfactory performance of the proposed model clearly. For small perturbations, the results have higher resolution comparing with the original MHD model; for large perturbations, the non-smooth solutions match well with existing references, confirming reliability of the model for instability investigations in magnetically confined plasma numerical research.

Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations

The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method.

A semi-implicit method for incompressible three-phase flow in porous media

In this paper, we present a semi-implicit method for the incompressible three-phase flow equations in two dimensions. In particular, a high-order discontinuous Galerkin spatial discretization is coupled with a backward Euler discretization in time. We consider a pressure-saturation formulation, decouple the pressure and saturation equations, and solve them sequentially while still keeping each equation implicit in its respective unknown. We present several numerical examples on both homogeneous and heterogeneous media, with varying permeability and porosity. Our results demonstrate the robustness of the scheme. In particular, no slope limiters are required and a relatively large time step may be taken.

A Stable 1D Multigroup High-Order Low-Order Method

ABSTRACTThe high-order low-order (HOLO) method is a recently developed moment-based acceleration scheme for solving time-dependent thermal radiative transfer problems, and has been shown to exhibit orders of magnitude speedups over traditional time-stepping schemes. However, a linear stability analysis by Haut et al. (2015) revealed that the current formulation of the multigroup HOLO method was unstable in certain parameter regions. Since then, we have replaced the intensity-weighted opacity in the first angular moment equation of the low-order (LO) system with the Rosseland opacity. This results in a modified HOLO method (HOLO-R) that is significantly more stable.This paper has two primary components. First, we present results from a linear stability analysis of the HOLO-R method for a 2-group problem. These results show that HOLO-R is stable in all regions of the parameter space we considered, and strongly suggest that HOLO-R is unconditionally stable. The predicted decay factors from the stability analysis are verified with numerically computed decay factors from a spectral HOLO-R code. Second, we describe a preliminary (unoptimized) implementation of HOLO-R in Capsaicin – a deterministic, discrete-ordinates radiation transport code at Los Alamos National Laboratory. The results from this implementation further verify the stability of the HOLO-R method. Moreover, in the simulation of a 50-group, nonlinear Marshak wave problem, away from the 2-group, linearized stability analysis, we assess HOLO-R’s accuracy, efficiency, and robustness as compared to current methods in Capsaicin. To our knowledge, this is the first implementation of HOLO that uses a mass-lumped discontinuous Galerkin spatial discretization for both the high-order (HO) and LO systems, and details of this discretization are provided in the Appendix.

Implicit Multistage Two-Derivative Discontinuous Galerkin Schemes for Viscous Conservation Laws

In this paper we apply implicit two-derivative multistage time integrators to conservation laws in one and two dimensions. The one dimensional solver discretizes space with the classical discontinuous Galerkin method, and the two dimensional solver uses a hybridized discontinuous Galerkin spatial discretization for efficiency. We propose methods that permit us to construct implicit solvers using each of these spatial discretizations, wherein a chief difficulty is how to handle the higher derivatives in time. The end result is that the multiderivative time integrator allows us to obtain high-order accuracy in time while keeping the number of implicit stages at a minimum. We show numerical results validating and comparing methods.

Shock capturing data assimilation algorithm for 1D shallow water equations

We propose a new data assimilation algorithm for shallow water equations in one dimension. The algorithm is based upon Discontinuous Galerkin spatial discretization of shallow water equations (DG-SW model) and the continuous formulation of the minimax filter. The latter allows for construction of a robust estimation of the state of the DG-SW model and computes worst-case bounds for the estimation error, provided the uncertain parameters belong to a given bounding set. Numerical studies show that, given sparse observations from numerical or physical experiments, the proposed algorithm quickly reconstructs the true solution even in the presence of shocks, rarefaction waves and unknown values of model parameters. The minimax filter is compared against the ensemble Kalman filter (EnKF) for a benchmark dam-break problem and the results show that the minimax filter converges faster to the true solution for sparse observations.

Large-Scale Three-Dimensional Acoustic Horn Optimization

We consider techniques that enable large-scale gradient-based shape optimization of wave-guiding devices in the context of three-dimensional time-domain simulations. The approach relies on a memory efficient boundary representation of the shape gradient together with primal and adjoint solvers semiautomatically generated by the FEniCS framework. The hyperbolic character of the governing linear wave equation, written as a first-order system, is exploited through systematic use of the characteristic decomposition both to define the objective function and to obtain stable numerical fluxes in the discontinuous Galerkin spatial discretization. The methodology is successfully used to optimize the shape of a midrange acoustic horn, described by 1,762 design variables, for maximum transmission efficiency, where the parallel computations involve a total of $3.5\times10^9$ unknowns.