Runge-Kutta Discontinuous Galerkin Finite Element Research Articles

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.

Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

A time-explicit Runge–Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection–reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ⩾ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results indicate that the proposed scheme is more robust and accurate in comparison to existing approaches based on the continuous Galerkin finite element method or the fast Fourier transform method.

The ratio of working memory and parameters of sensorimotor integration in the 'go-go' paradigm in children 7-8 years old

In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.

Application of discontinuous Galerkin method for solving a compressible five-equation two-phase flow model

Abstract In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.

Application of Runge-Kutta Discontinuous Galerkin finite element method to shallow water flow

Numerical solutions of 1D and 2D shallow water equations are presented with Runge-Kutta Discontinuous Galerkin (RKDG) finite element method. For 1D problems, the transcritical flows such as dam-break flows and a flow over a bump with hydraulic jump were simulated and the numerical solutions were compared with the exact solutions. As a formulation of approximate Riemann solver, the Local Lax-Wendroff (LLF) fluxes were employed and minmod slope limiter was used for 1D flows. For 2D applications, the classical problems of lateral transition were simulated. The HLL (Harten — Lax — van Leer) fluxes were adopted and a van Albada type gradient-reconstruction type slope limiter was applied. For the time integration, 3rd-order and 2nd-order TVD Runge-Kutta schemes were used for 1D and 2D simulations, respectively. In all case studies, good agreement was observed.

Local time-stepping in Runge–Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations

Large geophysical flows often encompass both coarse and highly resolved regions. Approximating these flows using shock-capturing methods with explicit time stepping gives rise to a Courant–Friedrichs–Lewy (CFL) time step constraint. Even if the refined regions are sparse, they can restrict the global CFL condition to very small time steps, vastly increasing computational effort over the whole domain. One method to cope with this problem is to use locally varying time steps over the domain. These are also referred to as multi-rate methods in the ODE literature. Ideally, such methods must be conservative, accurate and easy to implement. In this study, we derive a second-order, local time stepping procedure within a Runge–Kutta discontinuous Galerkin (RKDG) framework to solve the shallow water equations. This procedure is based on previous first-order work of the second author and collaborator Kirby [1–3]. As we are interested in both coastal and overland flows due to, e.g., rainfall, wetting and drying is incorporated into the model. Numerical results are shown, which verify the accuracy and efficiency of the approach (compared to using a globally defined CFL time step), and the application of the method to rainfall–runoff scenarios.

Runge-Kutta discontinuous Galerkin finite element method for two-dimensional gas dynamic equations in unified coordinate

In this paper we develop the Runge-Kutta discontinuous Galerkin finite element method for two-dimensional compressible gas dynamic equations in unified coordinate. The equations for fluid dynamics and geometry conservation laws are solved simultaneously. All the calculations can be performed on the fixed meshes. The information about the grid velocities is not needed in calculation. Some numerical examples are given to evaluate the efficiency and the reliability of the scheme. The numerical results show that the algorithm works well.

Discontinuous Galerkin Method for 1D Shallow Water Flows in Natural Rivers

A numerical model is proposed for the solution of one-dimensional shallow water flow equations for natural rivers. This model is based on the total variation diminishing Runge-Kutta discontinuous Galerkin finite element method. In natural rivers, the cross-section shape and bed slope can be quite irregular, which requires a compatible discretization scheme for the bed slope term and net pressure force term. Therefore, in this model, the hydrostatic pressure force term and the wall pressure force term are combined and a new discretization for the resulting term is introduced. This formulation is shown to prevent unphysical flow due to improper treatment of bottom slope term. The mass and momentum flux term are calculated by HLL Riemann solver. A scheme is presented to model flow over dry bed. To evaluate the numerical scheme, tests are conducted for idealized dambreak problems in parabolic (with wet and dry beds), and rectangular channels, hydraulic jump in a rectangular channel, dambreak in the Teton River (Idaho, USA) and the Toce River (Northern Alps, Italy), and flooding event in the East Fork River (Wyoming, USA). The comparison of the computational results with analytical and laboratory results of dam break flows shows that the model is capable of handling flow over dry areas. The simulation results for hydraulic jump show the discharge conservation property and shock prediction capability of the model. The dambreak and flood simulations in natural channels show that the model is capable of handling flows in highly varying bed topography and channel geometry.

Discontinuous Galerkin Method for 1D Shallow Water Flow in Nonrectangular and Nonprismatic Channels

A total variation diminishing Runge-Kutta discontinuous Galerkin finite element method for the solution of one-dimensional (1D) shallow water flow equations for natural channels is presented. The hydrostatic pressure force and the wall pressure force terms are combined to simplify the calculations and prevent unphysical flow attributable to improper treatment of the bottom slope term. The treatment of the combined term that appropriately accounts for the momentum flux is given. HLL and Roe Riemann solvers are assessed for the mass and momentum flux terms. Numerical tests are conducted using prismatic rectangular and nonrectangular channels as well as non prismatic channels and natural channel for dam break, supercritical flow, transcritical flow, and dry-bed problems. Slope limiters based on flow cross section area, water surface, and water depth are evaluated. The tests show that HLL and Roe solvers provide similar accuracy. However, the slope limiter based on flow area provides more accurate solutions f...

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

AbstractIn this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.

Quench Simulation in Cable-in-Conduit Conductor Using Runge-Kutta Discontinuous Galerkin Finite Element Method

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The need of high magnetic field facilities has raised the question of accurate analysis of stability and quench characteristics of forced-flow cooled cable-in-conduit conductor (CICC). The supercritical helium flow is considered in the model of quench propagation in CICC, which is described by Euler equations. In this paper, a scheme named Runge-Kutta discontinuous Galerkin finite element method, was introduced to simulate the quench propagation. We used Gauss-Legendre method to solve the integration in discreted equations, and Lax-Friedrichs fluxes were applied to replace the numerical fluxes at the boundary between elements. The simulated results were compared with experimental data, and they were in good agreement. </para>

Experimental and numerical investigations of dike-break induced flows

Experimental model data are compared with numerical computations of dike-break induced flows, focusing on the final steady state. An idealized scale model was designed reproducing the specific boundary conditions of dike breaks. Discharges, water level, and depth profiles of horizontal velocities were recorded and validated by numerical modeling. The latter was performed by two different models solving the two-dimensional depth-averaged shallow water equations, namely a total variation diminishing Runge-Kutta discontinuous Galerkin finite element method, and a finite volume scheme involving a flux vector splitting approach. The results confirmed convergence and general applicability of both methods for dike-break problems. As regards their accuracy, the basic flow pattern was satisfactorily reproduced yet with differences compared to the measurements. Hence, additional simulations by the finite volume model were performed considering various turbulence closures, wall-roughnesses as well as nonuniform Boussinesq coefficients.

Experimental and numerical investigations of dike-break induced flows

Experimental model data are compared with numerical computations of dike-break induced flows, focusing on the final steady state. An idealized scale model was designed reproducing the specific boundary conditions of dike breaks. Discharges, water level, and depth profiles of horizontal velocities were recorded and validated by numerical modeling. The latter was performed by two different models solving the two-dimensional depth-averaged shallow water equations, namely a total variation diminishing Runge-Kutta discontinuous Galerkin finite element method, and a finite volume scheme involving a flux vector splitting approach. The results confirmed convergence and general applicability of both methods for dike-break problems. As regards their accuracy, the basic flow pattern was satisfactorily reproduced yet with differences compared to the measurements. Hence, additional simulations by the finite volume model were performed considering various turbulence closures, wall-roughnesses as well as nonuniform Bous...

Numerical Simulation of Highly Underexpanded Axisymmetric Jet with Runge-Kutta Discontinuous Galerkin Finite Element Method

Highly underexpanded axisymmetric jet was simulated using the Runge-Kutta Discontinuous Galerkin (RKDG) finite element method, which, based on two-dimensional conservation laws, was used to solve the axisymmetric Euler equations. The computed results show that the complicated flow field structures of interest, including shock waves, slipstreams and the triple point observed in experiments could be well captured using the RKDG finite element method. Moreover, comparisons of the Mach disk location exhibit excellent agreements between the computed results and experimental measurements, indicating that this method has high capability of capturing shocks without numerical oscillation and artificial viscosity occurring near the discontinuous point.

The Runge-Kutta DG finite element method and the KFVS scheme for compressible flow simulations

This article presents a new type of second-order scheme for solving the system of Euler equations, which combines the Runge-Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method-of-lines approach will be discretized using a Runge-Kutta method. Finally, a second-order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one- and two-dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

RKDG Finite Element Method Combined with BGK Scheme for Solving Fluid Dynamics System

This paper presents a new type of second-order scheme for solving the system of Euler equations, which combines the Runge--Kutta discontinuous Galerkin (RKDG) finite element method and the Bhatnagar--Gross--Krook (BGK) scheme. We first discretize the Euler equations in space with the DG method, and then the resulting system will be discretized using an RK method. Finally, a second-order BGK method is used to construct the numerical flux. The proposed scheme keeps the main advantages of the DG finite element method, including its flexibility in handling irregular solution domains and in parallelization. Moreover, with the BGK approach the proposed schemes avoid using Riemann solvers. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one- and two-dimensional spaces.

Discontinuous Galerkin Finite-Element Method for Transcritical Two-Dimensional Shallow Water Flows

A total variation diminishing Runge Kutta discontinuous Galerkin finite-element method for two-dimensional depth-averaged shallow water equations has been developed. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally as efficient as comparable finite-volume schemes. For smooth parts of the solution, the scheme is second order for linear elements and third order for quadratic shape functions both in time and space. Shocks are usually captured within only two elements. Several steady transcritical and transient flows are investigated to confirm the accuracy and convergence of the scheme. The results show excellent agreement with analytical solutions. For investigating a flume experiment of supercritical open-channel flow, the method allows very good decoupling of the numerical and mathematical model, resulting in a nearly grid-independent solution. The simulation of an actual dam break shows the applicability of the scheme to nontrivial bathymetry and wave propagation on a dry bed.