Runge-Kutta Discontinuous Galerkin Method Research Articles

A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

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.

Wavefield simulation with the discontinuous Galerkin method for a poroelastic wave equation in triple-porosity media

In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biot’s theory and Lagrangian formulas, we obtain 3D Biot’s equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single- or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs flux and an implicit-weighted RK time discretization scheme are used for this computation. We use two types of mesh elements — quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P waves, P1 and P2 waves, in double-porosity media, whereas three kinds of slow P waves, P1, P2, and P3 waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation using the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multiporosity media computed by DGM. The slow P wave in multiporosity media with dissipation decays rapidly, which also will lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and indicate that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.

Numerical analysis on the high-speed impulsive noise propagation characteristic of helicopter rotor in the presence of strong shear flow

Based on the Linearized Euler Equations (LEE) and the Computational Fluid Dynamics (CFD) technique, the propagation characteristics of helicopter rotor High-Speed Impulsive (HSI) noise in non-uniform flowfield are simulated and analyzed. A Hybrid Computational Aeroacoustics (HCAA) method for solving the sound field of rotor is first developed. The acoustic source region is simulated by solving Navier-Stokes equations, and the acoustic near-field is simulated by solving LEE based on the Runge-Kutta Discontinuous Galerkin (RKDG) method. The numerical analysis method is then validated through comparisons with experimental data, and the aeroacoustics characteristics of the UH-1H rotor are calculated. Afterwards, the propagation characteristics of HSI noise in the wind tunnel are calculated, and the “trap” effect of the shear layer on HSI noise is determined. Finally, the peak negative pressure, time-averaged acoustic energy, and instantaneous perturbed pressure field are analyzed in detail, and some conclusions are drawn. In jet flow, the HSI noise energy is reinforced, and the time of the peak negative pressure is significantly advanced. In still air, the HSI noise energy is attenuated, and the time of the peak negative pressure is slightly delayed. These investigations offer valuable data for developing advanced theoretical correction method to consider the effect of shear layer on rotating acoustic source.

Modeling 3-D Elastic Wave Propagation in TI Media Using Discontinuous Galerkin Method on Tetrahedral Meshes

Transversely isotropic (TI) medium is a widely studied anisotropic solid medium in seismology. The numerical simulation of seismic wave propagation in TI media is an effective tool to analyze the mechanism of seismic waves in complex anisotropic media. In this study, we introduce a double-weighted Runge-Kutta discontinuous Galerkin (RKDG) method for numerically solving wave propagation problems in 3D TI media with surface topography. This method incorporates the discontinuous Galerkin spatial discretization with an explicit double-weighted two-step iteration time discretization. The local Lax-Friedrichs flux is used as the numerical flux in the formulations. This method can solve the first-order velocity-stress seismic wave equations including TI media as well as more general anisotropic media. Due to the large scale of 3D problems, the parallel technology is adopted. Regions with irregular boundaries are discretized into unstructured tetrahedral meshes. Numerical experiments for various anisotropic media including the vertical and tilted TI media are presented. The results demonstrate the effectiveness of the double-weighted RKDG method in wavefield simulations in 3D complicated anisotropic media.

Numerical study on the noise propagation characteristics of rotor in non-uniform downwash flowfield Based on Linearized Euler Equations

To study the influence of non-uniform flowfield on the propagation characteristics of helicopter rotor noise, a Hybrid Computational Aeroacoustics (HCAA) method is developed. The acoustic source region is simulated by Computational Fluid Dynamics (CFD) technique with the Unsteady Reynolds Averaged Navier-Stokes equations (URANS) as the governing equations. Acoustic near-field is simulated by Computational Aeroacoustics (CAA) technique with the Linearized Euler Equations (LEE) as the governing equations, and the numerical discretization of the LEE is accomplished by Runge-Kutta Discontinuous Galerkin (RKDG) method. A novel acoustic source extraction method based on pressure and pressure gradient is proposed to accomplish the one-way CFD-CAA weak coupling. The HCAA method is validated through comparisons with noise experimental data of the UH-1H model rotor and the BO-105 model rotor. Based on the proposed HCAA method, the convection and refraction effects of rotor noise under different collective pitch angles are analyzed. The results show that the distortion effect of the rotor noise is most affected by the non-uniformly distributed downwash velocity field, resulting in an increment of acoustic energy below the rotor plane. The effect of non-uniformly distributed downwash velocity on noise propagation increases with the increase of the collective pitch angle. For the UH-1H model rotor, the maximum change of the sound pressure level is 0.8 dB (about 10% change of the effective sound pressure).

Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous Galerkin method

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous Galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous Galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

High-Order CFD Solvers on Three-Dimensional Unstructured Meshes: Parallel Implementation of RKDG Method with WENO Limiter and Momentum Sources

In aerospace engineering, high-order computational fluid dynamics (CFD) solvers suitable for three-dimensional unstructured meshes are less developed than expected. The Runge–Kutta discontinuous Galerkin (RKDG) finite element method with compact weighted essentially non-oscillatory (WENO) limiters is one of the candidates, which might give high-order solutions on unstructured meshes. In this article, we provide an efficient parallel implementation of this method for simulating inviscid compressible flows. The implemented solvers are tested on lower-dimensional model problems and real three-dimensional engineering problems. Results of lower-dimensional problems validate the correctness and accuracy of these solvers. The capability of capturing complex flow structures even on coarse meshes is shown in the results of three-dimensional applications. For solving problems containing rotary wings, an unsteady momentum source model is incorporated into the solvers. Such a finite element/momentum source hybrid method eliminates the reliance on advanced mesh techniques, which might provide an efficient tool for studying rotorcraft aerodynamics.

An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes

In this paper, we improve the simple WENO limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method in [34] for solving two-dimensional hyperbolic systems on unstructured meshes. The major improvement is reducing the number of polynomials transformed to the characteristic fields for each direction. For the triangular mesh, this new simple WENO limiter transforms only two polynomials to the characteristic fields for each direction, while the original simple WENO limiter [34] uses four polynomials. Thus the improved simple WENO limiter reduces the computational cost and improves the efficiency. It provides a simpler and more practical and efficient way to the characteristic-wise limiting procedure, while still simultaneously maintaining uniform high-order accuracy in smooth regions and controlling spurious nonphysical oscillations near discontinuities. We also apply this improved simple WENO limiter to a high order method constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds [10], where polygonal meshes are constructed based on random points and the traditional DG method was adopted on the constructed polygonal mesh. For such a complex polygonal mesh, this limiter still transforms only two polynomials to the characteristic fields. Thus the simplicity and efficiency of this new limiter is more evident in this case. Numerical results are provided to illustrate the accuracy and effectiveness of this procedure. For some examples, the improved limiter has less smearing and higher resolution than the original one.

A well-balanced Runge-Kutta discontinuous Galerkin method for the Euler equations in isothermal hydrostatic state under gravitational field

This article presents a well-balanced Runge-Kutta discontinuous Galerkin method for the Euler equations with gravitation and in the isothermal hydrostatic state. To achieve the well-balanced feature, we develop a decomposition algorithm backed by a novel auxiliary function. Using the decomposition algorithm, we successfully build well-balanced numerical fluxes and achieve novel discretizations to the source term, then realize the well-balancedness ultimately. A series of examples show the well-balanced feature, the high-order accuracy, and the high resolution.

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-LobattoLegendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.

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.

Simulating compressible two-phase flows with sharp-interface discontinuous Galerkin methods based on ghost fluid method and cut cell scheme

A sharp-interface Runge-Kutta discontinuous Galerkin method with the combination of Ghost Fluid Method and Cut Cell approach is developed to simulate compressible two-medium flows with free interfaces. In this approach, the material interface is explicitly described by a simple cut line, and is implicitly evolved with Level Set method. Around the small cells which appear when the regular cells are passed by the interface, some ghost fluids are stored in these cells. The small cells are updated by an improved real-ghost mixing method to avoid the complicated cell-merge approach and the topological change difficulty encountered in our previous work. Under the DG frame, the improved mixing method can avoid the excessive numerical dissipation which is caused by the original mixing method presented in the previous work. The Runge-Kutta discontinuous Galerkin method is applied to calculate the flow field, and the HLL-type approximate Riemann solver is applied for the face flux. The adaptive mesh refinement technology is also adapted to reduce computing costs. Various numerical examples are tested to show the robustness and capability of the presented method.

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

A Mass Conservative Eulerian-Lagrangian Runge-Kutta Discontinuous Galerkin Method for Wave Equations with Large Time Stepping

A Mass Conservative Eulerian-Lagrangian Runge-Kutta Discontinuous Galerkin Method for Wave Equations with Large Time Stepping

A Compact Subcell WENO Limiting Strategy Using Immediate Neighbors for Runge-Kutta Discontinuous Galerkin Methods for Unstructured Meshes

In this paper, we generalize the compact subcell weighted essentially non oscillatory (CSWENO) limiting strategy for Runge-Kutta discontinuous Galerkin method developed recently in [1] for structured meshes to unstructured triangular meshes. The main idea of the limiting strategy is to divide the immediate neighbors of a given cell as subcells into the required stencil and to use a WENO reconstruction for limiting. This strategy can be applied for any type of WENO reconstruction. We have used the WENO reconstruction proposed in [2] and provided accuracy tests and results for two-dimensional Burgers’ equation and two dimensional Euler equations to illustrate the performance of this limiting strategy.

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

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation