Kutta Discontinuous Galerkin Method Research Articles

On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems

AbstractA posteriori bounds for the error measured in various norms for a standard second‐order explicit‐in‐time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one‐dimensional (in space) linear transport problem are derived. The proof is based on a novel space‐time polynomial reconstruction, hinging on high‐order temporal reconstructions for continuous and discontinuous Galerkin time‐stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.

An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier–Stokes equations

We consider an optimization-based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier–Stokes system is split into the compressible Euler equations, which are solved by the positivity-preserving Runge–Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, which is solved by Crank–Nicolson in time with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization-based limiter for the total energy variable to post-process DG polynomial cell averages. The optimization-based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas–Rachford splitting method by using nearly optimal algorithm parameters. Numerical tests suggest that the DG method with Qk basis and the optimization-based limiter is robust for demanding low-pressure problems such as high-speed flows.

The Runge–Kutta Discontinuous Galerkin Method with Compact Stencils for Hyperbolic Conservation Laws

The Runge–Kutta Discontinuous Galerkin Method with Compact Stencils for Hyperbolic Conservation Laws

A Conservative Eulerian–Lagrangian Runge–Kutta Discontinuous Galerkin Method for Linear Hyperbolic System with Large Time Stepping

A Conservative Eulerian–Lagrangian Runge–Kutta Discontinuous Galerkin Method for Linear Hyperbolic System with Large Time Stepping

A positivity-preserving implicit-explicit scheme with high order polynomial basis for compressible Navier–Stokes equations

In this paper, we are interested in constructing a scheme solving compressible Navier–Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., Δt=O(Δx). Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge–Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with Δt=O(Δx) if the Laplacian operator of internal energy is approximated by the Qk spectral element method with k=1,2,3. So the full scheme with Qk (k=1,2,3) basis is conservative and positivity-preserving with Δt=O(Δx), which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection.

Eulerian--Lagrangian Runge--Kutta Discontinuous Galerkin Method for Transport Simulations on Unstructured Meshes

Eulerian--Lagrangian Runge--Kutta Discontinuous Galerkin Method for Transport Simulations on Unstructured Meshes

L$^2$ Error Estimate to Smooth Solutions of High Order Runge--Kutta Discontinuous Galerkin Method for Scalar Nonlinear Conservation Laws with and without Sonic Points

In this paper we shall establish an a priori L$^2$-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant--Friedrichs--Lewy condition, and the quasi-optimal and/or optimal order of accuracy in space is achieved for many widely used numerical fluxes, no matter whether the solution contains sonic points or not. The main tools used in this paper are the matrix transferring process and the generalized Gauss--Radau projection of the reference functions, depending on the relative upwind effect. Finally some numerical experiments are given to support our theoretical results.

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

ABSTRACT This paper presents high-order Runge–Kutta (RK) discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components that are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane–Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples – including a toy model of stellar core collapse with a phenomenological equation of state that results in core bounce and shock formation – are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock-capturing capability, and total energy conservation.

An Efficient Parallel Implementation of the Runge–Kutta Discontinuous Galerkin Method with Weighted Essentially Non-Oscillatory Limiters on Three-Dimensional Unstructured Meshes

In computational fluid dynamics, high-order solvers suitable for three-dimensional unstructured meshes are attractive but are less developed than other methods. In this article, we provide the formulation and a parallel implementation of the Runge–Kutta discontinuous Galerkin finite element method with weighted essentially non-oscillatory limiters, which are compact and effective for suppressing numerical oscillations near discontinuities. In our experiments, high-order solvers do outperform their low-order counterparts in accuracy and the efficient parallel implementation makes the time cost affordable for large problems. Such high-order parallel solvers are efficient tools for solving conservative laws including the Euler system that models inviscid compressible flows.

An Efficient Discontinuous Galerkin Method Using a Tetrahedral Mesh for 3D Seismic Wave Modeling

ABSTRACTThe discontinuous Galerkin (DG) method is a numerical algorithm that is widely used in various fields. It has high accuracy and low numerical dispersion with advantages of easy handling boundary conditions and good parallel performance. In this study, we develop an efficient parallel weighted Runge–Kutta discontinuous Galerkin (WRKDG) method on unstructured meshes for solving 3D seismic wave equations. The DG method we use is based on the first-order formulation of a hyperbolic system with an explicit weighted Runge–Kutta time discretization. We describe the numerical scheme and parallel implementation in detail, and analyze the stability condition and numerical dispersion and dissipation. We carry out a convergence test on unstructured meshes and investigate the parallel efficiency of the implementation of the WRKDG method. The speedup curve shows that the method has good parallel performance. Finally, we present several numerical simulation examples, including acoustic and elastic wave propagations in isotropic and anisotropic media. Numerical results further verify the effectiveness of the WRKDG method in solving 3D wave propagation problems.

Local Error Estimates for Runge–Kutta Discontinuous Galerkin Methods with Upwind-Biased Numerical Fluxes for a Linear Hyperbolic Equation in One-Dimension with Discontinuous Initial Data

Local Error Estimates for Runge–Kutta Discontinuous Galerkin Methods with Upwind-Biased Numerical Fluxes for a Linear Hyperbolic Equation in One-Dimension with Discontinuous Initial Data

Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods

Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods

Runge–Kutta discontinuous Galerkin method for Baer–Nunziato model with ‘simple WENO’ limiting of conservative variables

Abstract The work is devoted to the application of Runge–Kutta discontinuous Galerkin (RKDG) method for solving Baer–Nunziato hyperbolic model for nonequilibrium two-phase flows. The approach is based on the application of the simple WENO limiter directly to the conservative variables. Mathematical model and the corresponding numerical algorithm are described. The results of numerical simulations for 1D and 2D tests are presented and discussed.

Thornado-hydro: A Discontinuous Galerkin Method for Supernova Hydrodynamics with Nuclear Equations of State*

Abstract This paper describes algorithms for nonrelativistic hydrodynamics in the toolkit for high-order neutrino radiation hydrodynamics (thornado), which is being developed for multiphysics simulations of core-collapse supernovae (CCSNe) and related problems with Runge–Kutta discontinuous Galerkin (RKDG) methods. More specifically, thornado employs a spectral-type nodal collocation approximation, and we have extended limiters—a slope limiter to prevent nonphysical oscillations and a bound-enforcing limiter to prevent nonphysical states—from the standard RKDG framework to be able to accommodate a tabulated nuclear equation of state (EoS). To demonstrate the efficacy of the algorithms with a nuclear EoS, we first present numerical results from basic test problems in idealized settings in one and two spatial dimensions, employing Cartesian, spherical-polar, and cylindrical coordinates. Then, we apply the RKDG method to the problem of adiabatic collapse, shock formation, and shock propagation in spherical symmetry, initiated with a 15 M ⊙ progenitor. We find that the extended limiters improve the fidelity and robustness of the RKDG method in idealized settings. The bound-enforcing limiter improves the robustness of the RKDG method in the adiabatic collapse application, while we find that slope limiting in characteristic fields is vulnerable to structures in the EoS—more specifically, in the phase transition from nuclei and nucleons to bulk nuclear matter. The success of these applications marks an important step toward applying RKDG methods to more realistic CCSN simulations with thornado in the future.

Simulation of shock-body interactions using the Runge–Kutta discontinuous Galerkin method with adaptive mesh refinement implemented on graphic accelerators

In this paper, a numerical scheme for solving the equations of the dynamics of a compressible fluid, based on the Runge–Kutta discontinuous Galerkin (RKDG) method and an adaptively refined cubic mesh (AMR), is applied to the problems of shock interaction with solid and liquid obstacles. Features of effective implementation algorithm for the graphic processors (GPU) are described. Several well-known validation test problems are considered. Results of full detailed three-dimensional simulations of the shock with the gas or fluid bubble using the developed GPU–RKDG–AMR solver are presented.

Study on the cavitation effects induced by the interaction between underwater blast and various boundaries

The load of cavitation collapse is one of the main input loads to the structure subjected to underwater explosion. The cavitation effects near different boundaries are of particular importance in understanding the load mechanisms and designing the protective structures. In present study, a numerical model that couples a Runge Kutta Discontinuous Galerkin method, a finite element method and an isentropic one-fluid cavitation model is developed to study the cavitation effects induced by underwater explosion near different boundaries. The boundaries of a rigid plate, a steel plate, a rubber coated plate and a foam coated plate are investigated in detail, after validating the coupled numerical model by a case of 1D cavitating flow in an open tube, an experiment of a water-backed plate subjected to water blast and an experiment of underwater explosion near a circular plate. The evolutions of the cavitation region, the pressure profile near the structure and the responses of the structure are comprehensively discussed. The results are beneficial for guiding the design of the protective structures.

The Influence of Various Structure Surface Boundary Conditions on Pressure Characteristics of Underwater Explosion

The shock wave of the underwater explosion can cause severe damage to the ship structure. The propagation characteristics of shock waves near the structure surface are complex, involving lots of complex phenomena such as reflection, transmission, diffraction, and cavitation. However, different structure surface boundaries have a significant effect on the propagation characteristics of pressure. This paper focuses on investigating the behavior of shock wave propagation and cavitation from underwater explosions near various structure surfaces. A coupled Runge–Kutta discontinuous Galerkin (RKDG) and finite element method (FEM) is utilized to solve the problem of the complex waves of fluids and structure dynamic response, considering the fluid compressibility. The level set (LS) method and the ghost fluid (GF) method are combined to capture the moving interface and deal with the stability of the coupling between the shock wave and structure surface. Besides, a cut-off cavitation model is introduced to the RKDG method. The validation of the numerical calculation model is discussed by comparing it with the known solution to verify the numerical solutions. Then, crucial kinds of structure surface boundary conditions include shallow-water single layer elasticity plate, double-layer crevasse elasticity plate, single layer curved elasticity plate, and double-layer curved elasticity plates are analyzed and discussed. The results and analysis can provide references for underwater explosion pressure characteristics, the impacting response of different boundary structures, and designing structures.

Superconvergence Analysis of the Runge–Kutta Discontinuous Galerkin Methods for a Linear Hyperbolic Equation

In this paper, we shall establish the superconvergence property of the Runge–Kutta discontinuous Galerkin (RKDG) method for solving a linear constant-coefficient hyperbolic equation. The RKDG method is made of the discontinuous Galerkin (DG) scheme with upwind-biased numerical fluxes coupled with the explicit Runge–Kutta algorithm of arbitrary orders and stages. Superconvergence results for the numerical flux, cell averages as well as the solution and derivative at some special points are shown, which are based on a systematical study of the $$\hbox {L}^2$$ -norm stability for the RKDG method and the incomplete correction techniques for the well-defined reference functions at each time stage. The result demonstrates that the superconvergence property of the semi-discrete DG method is preserved, and the optimal order in time is provided under the smoothness assumption that is independent of the number of stages. As a byproduct of the above superconvergence study, the expected order of the post-processed solution is obtained when a special initial solution is used. Some numerical experiments are also given.

Error Estimate of the Fourth-Order Runge--Kutta Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Error Estimate of the Fourth-Order Runge--Kutta Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Discontinuous Galerkin method for solving incompressible two-phase shallow granular flow model

This article deals with the numerical study of two-phase shallow flow model describing the mixture of fluid and solid granular particles. The model under investigation consists of coupled mass and momentum equations for solid granular material and fluid particles through non-conservative momentum exchange terms. The non-conservativity of model equations poses major challenges for any numerical scheme, such as well balancing, positivity preservation, accurate approximation of non-conservative terms, and achievement of steady-state conditions. Thus, in order to approximate the present model an accurate, well-balanced, robust, and efficient numerical scheme is required. For this purpose, in this article, Runge–Kutta discontinuous Galerkin method is applied successfully for the first time to solve the model equations. Several test problems are also carried out to check the performance and accuracy of our proposed numerical method. To compare the results, the same model is solved by staggered central Nessyahu–Tadmor scheme. A good comparison is found between two schemes, but the results obtained by Runge–Kutta discontinuous Galerkin scheme are found superior over the central Nessyahu–Tadmor scheme.