RKDG Method Research Articles

A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field

A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field

A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field

A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field

RKDG method solution for hyperbolic hyperelastic model

The natural for solid dynamics applications mathematical model has a form of the first-order hyperbolic system of conservation laws in the Eulerian reference frame. This model is proposed by Godunov and Romenski [1] and has a number of characteristic features from more traditional second-order formulations employed in elasticity. The most important is the presence of a complex structure of contact, shock and rarefaction waves and their interactions. Numerical solution of such problems is a difficult task due to a complex wave structure of the solution. Recently a number of efficient approaches were presented to handle a problem. Most of them are based on linearized HLL type solvers and high order WENO reconstruction techniques to obtain accurate fronts resolution in order to resolve complex solution structure. They also use Eulerian schemes based on solving Riemann problems for solid materials. In this paper we apply high order RKDG method as an alternative to WENO schemes to solve the problem. Up to 5-th order reconstruction is considered with 3-rd order timesteppnig scheme. The Riemann-free-solver approach is used with simple flux reconstruction. As a trial problem one dimensional hyperelastic formulation is presented similar to [1]. The developed computer code utilizes automatic differentiation (AD) approach which allows for an easy switch of underlying EOS and strain measure. The presented results cover discussion of the problem statement, corresponding RKDG approximations and simulation results that demonstrate possibilities of RKDG approach applied for the hyperbolic hyperelastic formulations in the Eulerian reference frame.

Сравнительное исследование лимитеров семейства WENO и Hermite WENO для расчета одномерных течений газа методом RKDG

Сравнительное исследование лимитеров семейства WENO и Hermite WENO для расчета одномерных течений газа методом RKDG

Effective solving of three-dimensional gas dynamics problems with the Runge-Kutta discontinuous Galerkin method

In this paper we present the Runge-Kutta discontinuous Galerkin method (RKDG method) for the numerical solution of the Euler equations of gas dynamics. The method is being tested on a series of Riemann problems in the one-dimensional case. For the implementation of the method in the three-dimensional case, a DiamondTorre algorithm is proposed. It belongs to the class of the locally recursive non-locally asynchronous algorithms (LRnLA). With the help of this algorithm a significant increase of speed of calculations is achieved. As an example of the three-dimensional computing, a problem of the interaction of a bubble with a shock wave is considered.

Providing the divergence-free property of the magnetic field when solving the MHD equations by the RKDG method

A review of methods providing the divergence-free property of the magnetic field when solving the two-dimensional problems of magnetic hydrodynamics on triangular grids by the secondorder discontinuous Galerkin method is presented. Procedures for redistributing the numerical flows corresponding to the magnetic field in the full two-dimensional formulation in the Cartesian coordinates and in the two-dimensional axisymmetric case in the cylindrical coordinates, making it possible to suppress the growth of the numerical divergence of the magnetic field, are described. Test problems of the Orszag–Tang vortex and spherical explosion are considered.

A discontinuous Galerkin method for modeling flow in networks of channels

Flow in a network of numerous channels is a complex phenomenon that cannot be adequately modeled by 1-D models. Despite this, it is a common practice to model flow in each channel with the 1-D St. Venant equations. These channels are coupled with each other by prescribing some interior boundary conditions, which are either too simplistic to reproduce the physics accurately or are computationally expensive. In this paper, we present a dimensionally heterogeneous approach to resolve these issues. We treat flows in individual channels as 1-D flows. These channels are coupled together at junctions using a 2-D junction region. We present a systematic numerical treatment of the 1-D and the 2-D shallow water equations in the framework of an RKDG method so that the coupling between the 1-D and the 2-D domains is natural, conservative and consistent with the numerical scheme. We verify the 1-D and the 2-D code via comparisons with different analytical solutions for a wide variety of cases. We establish the accuracy of our heterogeneous junction model by comparing the simulation results to experimental data and to the results obtained from other junction models.

DiamondTorre GPU Implementation Algorithm of the RKDG Solver for Fluid Dynamics and its Using for the Numerical Simulation of the Bubble-shock Interaction Problem

Abstract In this paper the solver based upon the RKDG method for solving three-dimensional Euler equations of gas dynamics is considered. For the numerical scheme the GPU implementation algorithm called DiamondTorre is used, which helps to improve the performance speed of cal- culations. The problem of the interaction of a spherical bubble with a planar shock wave is considered in the three-dimensional setting. The obtained calculations are in agreement with the known results of experiments and numerical simulations. The calculation results are ob- tained with the use of the PC.

Application of the RKDG method for gas dynamics problems

The Runge-Kutta discontinuous Galerkin method (RKDG method) is widely used nowadays. The RKDG method is characterized by a high-order accurate solution. As is well known, for ensuring the monotonic solution by this method, it is necessary to use slope limiters. In particular, it is necessary when the solution contains strong discontinuities. However, the application of limiters can negatively affect the accuracy of the solution. In this paper a way of saving the order of accuracy of the solution and ensuring its monotony is investigated.

The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two-dimensions

This paper presents a new cell-centered Lagrangian scheme for two-dimensional compressible flow. The new scheme uses a semi-Lagrangian form of the Euler equations. The system of equations is discretized by discontinuous Galerkin (DG) method using the Taylor basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently by a nodal solver. The mesh moves with the fluid flow. The time marching is implemented by a class of the Runge–Kutta (RK) methods. A WENO reconstruction is used as a limiter for the RKDG method. The scheme is conservative for the mass, momentum and total energy, and obeys the geometrical conservation law. The scheme maintains high-order accuracy and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.

A Runge Kutta Discontinuous Galerkin approach to solve reactive flows on conforming hybrid grids: the parabolic and source operators

A Runge–Kutta Discontinuous Galerkin method (RKDG) to solve the parabolic and source parts of reactive Navier–Stokes equations written in conservation form is presented. The parabolic operator uses a recent recovery method set up by van Leer for structured grids and a new projection method proposed by Borrel–Ryan for unstructured grids. The physical model involves complex chemistry and detailed transport. Transport coefficients are evaluated using algorithms which provide empirical expressions. In 1-D test cases the RKDG method is compared with a high order finite difference method. 2-D test cases in structured, unstructured and hybrid meshes are presented.

Application of the RKDG method for simulation of magneto-rotational instability

Application of the RKDG method for simulation of magneto-rotational instability

Numerical Simulation of Rayleigh-Taylor Instability with Large Density Ratios Based on RKDG Method

Rayleigh-Taylor instability problem with large density ratios is simulated by RKDG method which is developed for Euler equations with an additional body force corresponding to the gravity. The interface capturing ability of RKDG method is testified, while the density ratio (heavy to light) ranges from 3 to 20. Numerical results show that RKDG method has capability to pursue contact discontinuity in Rayleigh-Taylor instability with large density ratio. In the late stage of Rayleigh-Taylor instability problem, the contact line begins to crash, but the numerical solution is still smooth near the interface and has high resolution.

Применение RKDG метода второго порядка для решения двумерных уравнений идеальной магнитной гидродинамики

Применение RKDG метода второго порядка для решения двумерных уравнений идеальной магнитной гидродинамики

Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes

The shallow water equations are used to model large-scale surface flow in the ocean, coastal rivers, estuaries, salt marshes, bays, and channels. They can describe tidal flows as well as storm surges associated with extreme storm events, such as hurricanes. The resulting currents can transport bed load and suspended sediment and result in morphological changes to the seabed. Modeling these processes requires tightly coupling a bed morphology equation to the shallow water equations. Discontinuous Galerkin finite element methods are a natural choice for modeling this coupled system, given the need to solve these problems on unstructured computational meshes, as well as the desire to implement hp-adaptivity for capturing the dynamic features of the solution. However, because of the presence of non-conservative products in the momentum equations, the standard DG method cannot be applied in a straightforward manner. To rectify this situation, we summarize and follow an extended approach described by Rhebergen et al., which uses theoretical results due to Dal Maso et al. appearing in earlier work. In this paper, we focus on aspects of the implementation of the morphological model for bed evolution within the Advanced Circulation (ADCIRC) modeling framework, as well as the verification of the RKDG method in both h (mesh spacing) and p (polynomial order). This morphological model is applied to a number of coastal engineering problems, and numerical results are presented, with attention paid to the effects of h- and p-refinement in these applications. In particular, it is observed that for sediment transport, piecewise constant (i.e., finite volume) approximations of the bed are very over-diffusive and lead to poor sediment solutions.

Discontinuous Galerkin Finite-Element Method for Simulation of Flood in Crossroads

A numerical solution of the two-dimensional Saint Venant equations is presented for the study of the propagation of the floods through the crossroads of the city. The numerical scheme is a Runge-Kutta discontinuous Galerkin method (RKDG) with a slope limiter. The work studies the robustness and the stability of the method. The study is organized around three aspects: the prediction of the water depths, the location of the right and oblique hydraulic jumps in the crossing, and especially the distribution of the flow discharges in the downstream branches. The objective of this paper was to use the RKDG method in order to simulate supercritical flow in crossroads and to compare these simulations with experimental results and to show the advantage of this RKDG method compared to a second-order finite-volume method. A good agreement between the proposed method and the experimental data was found. The method is then able to simulate the flow patterns observed experimentally and to predict accurately the water depths, the location of the hydraulic jumps, and the discharge distribution in the downstream branches.

Computation of compressible flows with high density ratio and pressure ratio

The WENO method, RKDG method, RKDG method with original ghost fluid method, and RKDG method with modified ghost fluid method are applied to singlemedium and two-medium air-air, air-liquid compressible flows with high density and pressure ratios. We also provide a numerical comparison and analysis for the above methods. Numerical results show that, compared with the other methods, the RKDG method with modified ghost fluid method can obtain high resolution results and the correct position of the shock, and the computed solutions are converged to the physical solutions as the mesh is refined.

The RKDG Method and Its Application for the Numerical Solution of Gas Dynamics Problems

The RKDG Method and Its Application for the Numerical Solution of Gas Dynamics Problems

High-order RKDG Methods for Computational Electromagnetics

In this paper we introduce a new RKDG method for problems of wave propagation that achieves full high-order convergence in time and space. The novelty of the method resides in the way in which it marches in time. It uses an mth-order m-stage, low storage SSP-RK scheme which is an extension to a class of non-autonomous linear systems of a recently designed method for autonomous linear systems. This extension allows for a high-order accurate treatment of the inhomogeneous, time-dependent terms that enter the semi-discrete problem on account of the physical boundary conditions. Thus, if polynomials of degree k are used in the space discretization, the RKDG method is of overall order m = k + 1, for any k > 0. Moreover, we also show that the attainment of high-order space--time accuracy allows for an efficient implementation of post-processing techniques that can double the convergence order. We explore this issue in a one-dimensional setting and show that the superconvergence of fluxes previously observed in full space--time DG formulations is also attained in our new RKDG scheme. This allows for the construction of higher-order solutions via local interpolating polynomials. Indeed, if polynomials of degree k are used in the space discretization together with a time-marching method of order 2k + 1, a post-processed approximation of order 2k + 1 is obtained. Numerical results in one and two space dimensions are presented that confirm the predicted convergence properties.

High-order RKDG methods for computational electromagnetics

In this paper we introduce a new RKDG method for problems of wave propagation that achieves full high-order convergence in time and space. The novelty of the method resides in the way in which it m...