Discontinuous Galerkin Solution Research Articles

[formula omitted]-adaptive discontinuous Galerkin solution of transonic viscous flows with variable time step-size

The discontinuous Galerkin methods demonstrated to be well suited for scale resolving simulations of complex configurations, characterized by different fluid dynamics phenomena, such as transition, separation, shock/boundary layer interaction. Unfortunately, solvers based on these methods cannot yet reach the computational efficiency of well-established standard solvers, e.g., based on finite volume methods. To reduce the computational cost and the memory footprint, while not spoiling the accuracy, different approaches were investigated, which act on the spatial or the temporal discretization, and on the linear algebra. The above approaches have been extensively investigated, but rarely considering their mutual interactions.In this work, a p-adaptation algorithm is coupled with a time step-size adaptation algorithm to mitigate robustness issues arising after each adaptation cycle, probably related to the projection of the old solution on the new polynomial orders distribution. Moreover, this strategy is able to control the transient phase before each adaptation cycle, and can handle also the presence of initial steady solution, when the mesh is too coarse and the polynomial degree has not yet compensated for the lack of spatial accuracy. The effectiveness and accuracy of the proposed algorithm are demonstrated on transonic viscous flows with shock-wave/boundary-layer interaction.

Steady-state simulation of Euler equations by the discontinuous Galerkin method with the hybrid limiter

In the realm of steady-state solutions of Euler equations, the challenge of achieving convergence of residue close to machine zero has long plagued high-order shock-capturing schemes, particularly in the presence of strong shock waves. In response to this issue, this paper presents a hybrid limiter designed specifically for the discontinuous Galerkin (DG) method which incorporates a pseudo-time marching method. This hybrid limiter, initially introduced in DG methods for unsteady problems, preserves the local data structure while enhancing resolution through effective and precise shock capturing. Notably, for the steady problem, the hybridization of the DG solution with the cell average is employed, deviating from the low-order limited DG solution typically used for unsteady problems. This approach seamlessly integrates the previously distinct troubled cell indicator and limiter components resulting in a more cohesive and efficient limiter. It obviates the need for characteristic decomposition and intercell communication, leading to substantial reductions in computational costs and enhancement in parallel efficiency. Numerical experiments are presented to demonstrate the robust performance of the hybrid limiter for steady-state Euler equations both on the structured and unstructured meshes.

Discontinuous Galerkin Method of the First Order Parabolic Differential Equation

This paper is concerned with the mathematical study of the discontinuous Galerkin (DG) finite element method for the parabolic differential equation. The DG method is a vital numerical method with much mass compensation and more flexible meshing than other methods. In this study, we give a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The error analysis of this problem is also established. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov–Maxwell equations

Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov–Maxwell equations

Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

Standard discontinuous Galerkin finite element solutions to convection-dominated convection–diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem.

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.

Shallow-Flow Velocity Predictions Using Discontinuous Galerkin Solutions

Shallow-Flow Velocity Predictions Using Discontinuous Galerkin Solutions

Error analysis of a novel discontinuous Galerkin method for the two-dimensional Poisson's equation

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation uxx+uyy=f(x,y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x-spatial variable leading to a system of ordinary differential equations (ODEs) in the y-variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p-th and q-th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L2-norm of O(hp+1)+O(kq+1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments.

Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme

The present study aims to apply an implicit high-order reconstructed discontinuous Galerkin (DG) scheme (rDG(P 1 P 2)) to simulate the steady-state solar corona. In this scheme, a piecewise quadratic polynomial solution, P 2, is obtained from the underlying piecewise linear DG solution, P 1, by least-squares reconstruction with a weighted essentially nonoscillatory limiter. The reconstructed quadratic polynomial solution is then used for the computation of the fluxes and source terms. In addition, an implicit time integration method with large time steps is considered in this work. The resulting large linear algebraic system of equations from the implicit discretization is solved by the cellwise relaxation implicit scheme which can make full use of the compactness of the DG scheme. The code of the implicit high-order rDG scheme is developed in Fortran language with message passing interface parallelization in Cartesian coordinates. To validate this code, we first test a problem with an exact solution, which confirms the expected third-order accuracy. Then we simulate the solar corona for Carrington rotations 2167, 2183, and 2210, and compare the modeled results with observations. We find that the numerical results basically reproduce the large-scale observed structures of the solar corona, such as coronal holes, helmet streamers, pseudostreamers, and high- and low-speed streams, which demonstrates the capability of the developed scheme.

Hybrid discontinuous Galerkin-finite volume techniques for compressible flows on unstructured meshes

In this paper we develop a family of arbitrarily high-order non-oscillatory hybrid Discontinuous Galerkin(DG)-Finite Volume(FV) schemes for mixed-element unstructured meshes. Their key ingredient is a switch between a DG method and a FV method based on the CWENOZ scheme when invalid solutions are detected by a troubled cell indicator checking the unlimited DG solution. Therefore, the high order of accuracy offered by DG is preserved in smooth regions of the computational domain, while the robustness of FV is utilized in regions with strong gradients. The high-order CWENOZ variant used has the same spatial order of accuracy as the DG variant, while representing one of the most compact applications on unstructured meshes, therefore simplifying the implementation, reducing the computational overhead associated with large stencils of the original WENO reconstruction without sacrificing the desirable non-oscillatory properties of the schemes. We carefully investigate several parameters associated with the switching between DG and FV methods including the troubled cell indicators in a priori fashion. For the first time in the literature, we investigate the definition of the bounds for an admissible solution, the frequency by which we use the troubled cell indicators, and the evolution of the percentage of troubled cells for unsteady test problems. The 2D and 3D Euler equations are solved for well established test problems and compared with computational or experimental reference solutions. All the methods have been implemented and deployed within the UCNS3D open-source high-order unstructured Computational Fluid Dynamics (CFD) solver. The present coupling has the potential to improve the shortcomings of both FV-DG in a computational efficient manner. The improved accuracy and robustness provided is a characteristic of paramount importance for industrial-scale CFD applications, and favours the extension to other systems of governing equations.

Error profile for discontinuous Galerkin time stepping of parabolic PDEs

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate these results with some numerical experiments.

A reconstructed discontinuous Galerkin method based on variational formulation for compressible flows

A new reconstructed discontinuous Galerkin (rDG) method based on variational formulation is developed for compressible flows. In the presented method, a higher-order piece-wise polynomial is reconstructed based on the underlying discontinuous Galerkin (DG) solution. This reconstruction is done by using a newly developed variational formulation. The variational reconstruction (VR) can be seen as an extension of the compact finite difference (FD) schemes to unstructured grids. The reconstructed variables are obtained by solving an extreme-value problem, which minimizes the jumps of the reconstructed piece-wise polynomials across the cell interfaces, and therefore maximizes the smoothness of the reconstructed solution. Intrinsically, the stencils of the presented reconstruction are the entire mesh, so this method is robust even on tetrahedral grids. A variety of benchmark test cases are presented to assess the accuracy, efficiency and robustness of this rDG method. The numerical experiments demonstrate that the developed rDG method based on variational formulation can maintain the linear stability, obtain the designed high-order accuracy, and outperform the rDG counterpart based on the least-squares reconstruction for both inviscid and viscous compressible flows.

On a technique for reducing spurious oscillations in DG solutions of convection–diffusion equations

This note studies a generalization of a post-processing technique and a novel method inspired by the same technique which significantly reduce spurious oscillations in discontinuous Galerkin solutions of convection–diffusion equations in the convection-dominated regime.

Divergence Error Based p-adaptive Discontinuous Galerkin Solution of Time-domain Maxwell's Equations

A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive method. The nature of divergence error is explored and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operators of varying accuracy to substantial regions in the computational domain. This results in a reduced computational cost than a comparable discontinuous Galerkin time-domain solution using uniform degree piecewise polynomial bases throughout.

High order modal Discontinuous Galerkin Implicit–Explicit Runge Kutta and Linear Multistep schemes for the Boltzmann model on general polygonal meshes

Deterministic solutions of the Boltzmann equation represent a real challenge due to the enormous computational effort which is required to produce such simulations and often stochastic methods such as Direct Simulation Monte Carlo (DSMC) are used instead due to their lower computational cost. In this work, we show that combining different technologies for the discretization of the velocity space and of the physical space coupled with suitable time integration techniques, it is possible to compute very precise deterministic approximate solutions of the Boltzmann model in different regimes, from extremely rarefied to dense fluids, with CFL conditions only driven by the hyperbolic transport term. To that aim, we develop modal Discontinuous Galerkin (DG) Implicit–Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit–Explicit Linear Multistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF) for solving the Boltzmann model on multidimensional unstructured meshes. The solution of the Boltzmann collision operator is obtained through fast spectral methods, while the transport term in the governing equations is discretized relying on an explicit shock-capturing DG method on polygonal tessellations in the physical space. A novel class of WENO-type limiters, based on a shifting of the moments of inertia for each zone of the mesh, is used to control spurious oscillations of the DG solution across discontinuities. The use of Linear Multistep Methods (LMM) allows the Boltzmann solutions to be consistent not only with the compressible Euler limit but also with the Navier–Stokes asymptotic regime. In addition, as numerically proven, they also permit to strongly reduce the computational effort compared to Runge–Kutta approaches while maintaining the same or even larger accuracy. The performances of these different time discretization techniques are measured comparing both precision and efficiency. At the same time, comparisons against simpler relaxation type kinetic models such as the BGK model are proposed. The order of convergence is numerically measured for different regimes and found to agree with the theoretical findings. The new methods are validated considering two-dimensional benchmark test cases typically used in the fluid dynamics community. A prototype engineering problem consisting of a supersonic flow around a NACA 0012 airfoil with space–time-dependent boundary conditions is also presented for which the pressure coefficients are measured.

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

Abstract The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

The p-Weighted Limiter for the Discontinuous Galerkin Method in Solving Compressible Flows on Tetrahedral Grids

This paper makes several improvements to generalise the p-weighted limiter for the discontinuous Galerkin (DG) method proposed in 1D and triangular grids [Li, Wanai, Qian Wang, and Yu Xin Ren. 2020. “A P-weighted Limiter for the Discontinuous Galerkin Method on One-dimensional and Two-dimensional Triangular Grids.” Journal of Computational Physics 407: 109246] to tetrahedral grids for solving compressible flows. First, the limiter would be performed using the local coordinates instead of the physical coordinates, which coincides with the DG solution and saves the expensive cost in transforming the solution to the physical space. Second, the weighted procedure in the local coordinates can be more efficient by storing some cell-independent constants. Third, the parameter, which prevents very large weights on the linear candidate polynomials at extrema, is replaced by a formula to meet the requirement of ϵ. These improvements construct an efficient way in calculating compressible flows in complex geometries. Inviscid and viscous tests are calculated to validate the shock capturing capability of the limiter and the high-order accuracy of the DG schemes.

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Higher order accuracy is one of the well-known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. The smoothness-increasing accuracy-conserving (SIAC) filter is a popular post-processing technique introduced by Cockburn et al. (Math. Comput. 72(242): 577–606, 2003). It can raise the convergence rate of the DG solution (with a polynomial of degree k) from order \(k+1\) to order \(2k+1\) in the \(L^2\) norm. This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction. The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features, such as extra smoothness. Second, we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We prove the superconvergence error estimate of the new SIAC filters. Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.

Explicit high-order exponential time integrator for discontinuous Galerkin solution of Maxwell's equations

In this paper, exponential propagator (EP) is introduced to the discontinuous Galerkin finite element method (DG-FEM) for solving transient electromagnetic problems. As a kind of explicit time-stepping method, the proposed EP has fourth order accuracy in time and requires only one storage unit for each of the degrees-of-freedom. It provides an attractive alternative to the high-order low-storage explicit Runge-Kutta (LSERK) approach, which is the most popular time-marching scheme currently used in high-fidelity simulation of Maxwell's equations with DG-FEM. When centered numerical flux is utilized, the discretized Maxwell's system in lossless region is symplectic structure-preserving (energy conserving), and thus very suitable for accurately computing the long-time propagation of waves in non-dissipative medium, or simulating the motion of charged particles in electromagnetic fields. Our method can also handle upwind numerical flux at the sacrifice of symplectic-conserving virtue for higher accuracy in spatial discretization and spurious inhibition. A major contribution of this work is that, the application range of this scheme is extended to lossy media such as conductors and particularly unsplit perfectly matched layers in the framework of DG-FEM. This greatly facilitates the treatment of open boundary conditions in scattering and/or radiation problems. Extensive numerical examples, including the analysis of photonic crystals, dielectric coated perfect conducting cylinder, propagation of waves in zero-index materials, and corrugated O-type Cherenkov oscillators etc., are presented to demonstrate its stability, accuracy, and performance. All the numerical experiments show that the presented method is superior to the LSERK scheme when centered numerical flux is adopted.

Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations

We propose and analyze a class of robust, uniformly high-order accurate discontinuous Galerkin (DG) schemes for multidimensional relativistic magnetohydrodynamics (RMHD) on general meshes. A distinct feature of the schemes is their physical-constraint-preserving (PCP) property, i.e., they are proven to preserve the subluminal constraint on the fluid velocity and the positivity of density, pressure, and internal energy. This is the first time that provably PCP high-order schemes are achieved for multidimensional RMHD. Developing PCP high-order schemes for RMHD is highly desirable but remains a challenging task, especially in the multidimensional cases, due to the inherent strong nonlinearity in the constraints and the effect of the magnetic divergence-free condition. Inspired by some crucial observations at the PDE level, we construct the provably PCP schemes by using the locally divergence-free DG schemes of the recently proposed symmetrizable RMHD equations as the base schemes, a limiting technique to enforce the PCP property of the DG solutions, and the strong-stability-preserving methods for time discretization. We rigorously prove the PCP property by using a novel “quasi-linearization” approach to handle the highly nonlinear physical constraints, technical splitting to offset the influence of divergence error, and sophisticated estimates to analyze the beneficial effect of the additional source term in the symmetrizable RMHD system. Several two-dimensional numerical examples are provided to further confirm the PCP property and to demonstrate the accuracy, effectiveness and robustness of the proposed PCP schemes.