Runge-Kutta Time Discretization Research Articles

Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases.

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper, we analyze the explicit Runge-Kutta discontinuous Galerkin (RKDG) methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta (TVDRK3) time discretization and upwinding numerical fluxes. By using the energy method, under a standard Courant-Friedrichs-Lewy (CFL) condition, we obtain L 2 stability for general solutions and a priori error estimates when the solutions are smooth enough. The theoretical results are proved for piecewise polynomials with any degree k ⩾ 1. Finally, since the solutions to this system are non-negative, we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy. Numerical results are provided to demonstrate these RKDG methods.

Hermite WENO schemes for Hamilton–Jacobi equations on unstructured meshes

In this paper, we extend a class of the Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton–Jacobi equations by Qiu and Shu (2005) [24] to two dimensional unstructured meshes. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first two derivative values are evolved via time advancing and used in the reconstructions, while only the function values are evolved and used in the original WENO schemes which are nodal based approximations. The third and fourth order HWENO schemes using the combinations of second order approximations with nonlinear weights and TVD Runge–Kutta time discretization method are used here. Comparing with the original WENO schemes for Hamilton–Jacobi equations, one major advantage of HWENO schemes presented here is its compactness in the reconstructions. Extensive numerical tests are performed to illustrate the capability and high order accuracy of the methodologies.

Effect of one-way clutch on the nonlinear vibration of belt-drive systems with a continuous belt model

This study focuses on the nonlinear steady-state response of a belt-drive system with a one-way clutch. A dynamic model is established to describe the rotations of the driving pulley, the driven pulley, and the accessory shaft. Moreover, the model considers the transverse vibration of the translating belt spans for the first time in belt-drive systems coupled with a one-way clutch. The excitation of the belt-drive system is derived from periodic fluctuation of the driving pulley. In automotive systems, this kind of fluctuation is induced by the engine firing harmonic pulsations. The derived coupled discrete–continuous nonlinear equations consist of integro-partial-differential equations and piece-wise ordinary differential equations. Using the Galerkin truncation, a set of nonlinear ordinary differential equations is obtained from the integro-partial-differential equations. Applying the Runge–Kutta time discretization, the time histories of the dynamic response are numerically solved for the driven pulley and the accessory shaft and the translating belt spans. The resonance areas of the coupled belt-drive system are determined using the frequency sweep. The effects of the one-way clutch on the belt-drive system are studied by comparing the frequency–response curves of the translating belt with and without one-way clutch device. Furthermore, the results of 2-term and 4-term Galerkin truncation are compared to determine the numerical convergence. Moreover, parametric studies are conducted to understand the effects of the system parameters on the nonlinear steady-state response. It is concluded that one-way clutch not only decreases the resonance amplitude of the driven pulley and shaft's rotational vibration, but also reduces the resonance region of the belt's transverse vibration.

PERIODIC AND CHAOTIC RESPONSES OF AN AXIALLY ACCELERATING VISCOELASTIC BEAM UNDER TWO-FREQUENCY EXCITATIONS

This study focuses on the steady-state periodic response and the chaotic behavior in the transverse motion of an axially moving viscoelastic tensioned beam with two-frequency excitations. The two-frequency excitations come from the external harmonic excitation and the parametric excitation from harmonic fluctuations of the moving speed. A dynamic model is established to include the finite axial support rigidity, the material derivative in the viscoelastic constitution relation, and the longitudinally varying tension due to the axial acceleration. The derived nonlinear integro-partial-differential equation of motion possesses space-dependent coefficients. Applying the differential quadrature method (DQM) and the integral quadrature method (IQM) to the equation of the transverse motion, a set of nonlinear ordinary differential equations is obtained. Based on the Runge–Kutta time discretization, the time history of the axially moving beam is numerically solved for the case of the primary resonance, the super–harmonic resonance, and the principal parametric resonance. For the first time, the nonlinear dynamics is studied under various relations between the forcing frequency and the parametric frequency, such as equal, multiple, and incommensurable relationships. The stable periodic response and its sensitivity to initial conditions are determined using the bidirectional frequency sweep. Furthermore, chaotic motions are identified using different methods including the Poincaré map, the maximum Lyapunov exponent, the fast Fourier transforms, and the initial value sensitivity. Numerical simulations reveal the characteristics of the periodic, quasiperiodic, and chaotic motion of a nonlinear axially moving beam under two-frequency excitations.

The Method of Lines Solution of the Regularized Long-Wave Equation Using Runge-Kutta Time Discretization Method

A method of lines approach to the numerical solution of nonlinear wave equations typified by the regularized long wave (RLW) is presented. The method developed uses a finite differences discretization to the space. Solution of the resulting system was obtained by applying fourth Runge-Kutta time discretization method. Using Von Neumann stability analysis, it is shown that the proposed method is marginally stable. To test the accuracy of the method some numerical experiments on test problems are presented. Test problems including solitary wave motion, two-solitary wave interaction, and the temporal evaluation of a Maxwellian initial pulse are studied. The accuracy of the present method is tested with <svg style="vertical-align:-3.3907pt;width:22.5875px;" id="M1" height="15.4" version="1.1" viewBox="0 0 22.5875 15.4" width="22.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g> </svg> and <svg style="vertical-align:-3.27605pt;width:16.1625px;" id="M2" height="15.25" version="1.1" viewBox="0 0 16.1625 15.25" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"/></g> <g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg> error norms and the conservation properties of mass, energy, and momentum under the RLW equation.

A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan Mode

As an example of a front propagation, we study the propagation of a three-dimensional nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws, and it is termed a geometric solenoidal constraint. The analysis of a Cauchy problem for the linearized system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL-type reconstructions and Runge–Kutta time discretizations. A constrained transport–type technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented, which confirm the efficiency and robustness of the proposed numerical method and the control of the Jordan mode.

Runge-Kutta time discretization of parabolic differential equations on evolving surfaces

Runge-Kutta time discretization of parabolic differential equations on evolving surfaces

CENTRAL SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS

The semi-discrete central scheme and central upwind scheme use Runge-Kutta (RK) time discretization. We do the Lax-Wendroff (LW) type time discretization for both schemes. We perform numerical experiments for various problems including two dimensional Riemann problems for Burgers' equation and Euler equations. The results show that the LW time discretization is more efficient in CPU time than the RK time discretization while maintaining the same order of accuracy.

High-order, finite-volume methods in mapped coordinates

We present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourth-order accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge–Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.

Simulations of Shallow Water Equations with Finite Difference Lax-Wendroff Weighted Essentially Non-oscillatory Schemes

In this paper we study a Lax-Wendroff-type time discretization procedure for the finite difference weighted essentially non-oscillatory (WENO) schemes to solve one-dimensional and two-dimensional shallow water equations with source terms. In order to maintain genuinely high order accuracy and suit to problems with a rapidly varying bottom topography we use WENO reconstruction not only to the flux but also to the source terms of algebraical modified shallow water equations. Extensive simulations are performed, as a result, the WENO schemes with Lax-Wendroff-type time discretization can maintain nonoscillatory properties and more cost effective than that with Runge-Kutta time discretization.

High-Order Computational Scheme for a Dynamic Continuum Model for Bi-Directional Pedestrian Flows

In this article, the authors present a high-order weighted essentially non-oscillatory (WENO) scheme, coupled with a high-order fast sweeping method, for solving a dynamic continuum model for bi-directional pedestrian flows. The dynamic continuum model for bi-directional pedestrian flows is reviewed first, which is composed of a coupled system of a conservation law and an Eikonal equation. Next, the authors present the first-order Lax–Friedrichs difference scheme with first-order Euler forward time discretization, the third-order WENO scheme with third-order total variation diminishing (TVD) Runge–Kutta time discretization, and the fast sweeping method, and demonstrate how to apply them to the model under study. A comparison of the numerical results of the model from the first-order and high-order methods is provided, and it is concluded that the high-order method is more efficient than the first-order one, and they both converge to the same solution of the physical model.

A Numerical Study for the Performance of the WENO Schemes Based on Different Numerical Fluxes for the Shallow Water Equations

In this paper we investigate the performance of the weighted essential non-oscillatory (WENO) methods based on difierent numerical ∞uxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical ∞uxes. We consider six numerical ∞uxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the flrst-order centered ∞uxes, with the WENO flnite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities.

Level Set interface treatment and its application in Euler method

Level Set interface treatment method is introduced into Euler method, which is employed for interface treatment method for multi-materials. Combined with the ghost fluid method, the moving interface is tracked. Fifth-order WENO spatial discretization and third-order TVD Runge-Kutta time discretization methods are used. Shock-wave action on bubble, implosion and velocity field Shock effect bubbles; implosion and velocity field are simulated by means of LS-MMIC3D programmed by C++. Numerical results show that the Level Set interface treatment method is effective and feasible for multi-material interface treatment in comparison with the WENO method.

A Genuinely Multi-Dimensional Relaxed Upwind Scheme for Nonlinear Hyperbolic Problems

In this paper we present the results of a genuinely multi-dimensional relaxed upwind scheme for an arbitrary system of conservation laws in two space dimensions. A new discrete velocity Boltzmann equation is proposed, which is an improvement over the previously introduced models in terms of the isotropic coverage of the multi-dimensional domain by the foot of the characteristic. A finite volume method has been developed in which the fluxes at the cell interfaces are evaluated in a genuinely multi-dimensional way, in contrast to the traditional dimension by dimension treatment. Second order accuracy of the scheme is achieved by using standard MUSCL type reconstructions and TVD Runge-Kutta time discretizations. The results of several numerical experiments on the compressible Euler equations are presented which confirm the robustness and correct multi-dimensional behaviour of the new scheme.

A discontinuous Galerkin method for viscous compressible multifluids

We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier–Stokes system of equations. The system presented has a functional viscosity ν which depends on the pressure p=p(ρ,μi) of the flow, with the density ρ and the local concentration μi. High order Runge–Kutta time-discretization techniques are employed, and different methods of dealing with arbitrary coupled boundary conditions are discussed. Analysis of the energy consistency of the scheme is performed in addition to inspection of the relative error of the solution compared to exact analytic test cases. Finally several examples, comparisons, generalizations and physical applications are presented.

High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations

In this paper, we explore the Lax–Wendroff (LW) type time discretization as an alternative procedure to the high order Runge–Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in [3,5]. The LW time discretization is based on a Taylor expansion in time, coupled with a local Cauchy–Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial discretization using quadrilateral meshes [3] and third-order spatial discretization using curvilinear meshes [5]. Comparing with the Runge–Kutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.

Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics

The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. The methods are specially suited for efficient implementation on structured meshes. The hyperbolic system is required to be non-stiff. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas. Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We have shown that the most elegant and compact formulation of WENO reconstruction obtains when the interpolating functions are expressed in modal space. Explicit formulae have been provided for schemes having up to fourth order of spatial accuracy. Divergence-free evolution of magnetic fields requires the magnetic field components and their moments to be defined in the zone faces. We draw on a reconstruction strategy developed recently by the first author to show that a high order specification of the magnetic field components in zone-faces naturally furnishes an appropriately high order representation of the magnetic field within the zone. We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space–time Galerkin formulation instead of the usual Cauchy–Kovalewski procedure. We call such schemes ADER-CG and show that a very elegant and compact formulation results when the scheme is formulated in modal space. Explicit formulae have been provided on structured meshes for ADER-CG schemes in three dimensions for all orders of accuracy that extend up to fourth order. Such ADER schemes have been used to temporally evolve the WENO-based spatial reconstruction. The resulting ADER-WENO schemes provide temporal accuracy that matches the spatial accuracy of the underlying WENO reconstruction. In this paper we have also provided a point-wise description of ADER-WENO schemes for divergence-free MHD in a fashion that facilitates computer implementation. The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. All the methods presented have a one-step update, making them low-storage alternatives to the usual Runge–Kutta time-discretization. Their one-step update also makes them suitable building blocks for adaptive mesh refinement (AMR) calculations. We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions. It is shown that the increasing computational complexity with increasing order is handily offset by the increased accuracy of the scheme. The resulting ADER-WENO schemes are, therefore, very worthy alternatives to the standard second order schemes for compressible Euler and MHD flow.

Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We derive CFL conditions for the linear stability of the so-called Runge–Kutta discontinuous Galerkin (RKDG) methods on triangular grids. Semidiscrete DG approximations using polynomials spaces of degree p = 0 , 1 , 2 , and 3 are considered and discretized in time using a number of different strong-stability-preserving (SSP) Runge–Kutta time discretization methods. Two structured triangular grid configurations are analyzed for wave propagation in different directions. Approximate relations between the two-dimensional CFL conditions derived here and previously established one-dimensional conditions can be observed after defining an appropriate triangular grid parameter h and a constant that is dependent on the polynomial degree p of the DG spatial approximation. Numerical results verify the CFL conditions that are obtained, and “optimal”, in terms of computational efficiency, two-dimensional RKDG methods of a given order are identified.

A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes

In this paper, we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes based on the work (Computers & Fluids, 34: 642–663 (2005)) by Qiu and Shu, with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws. The key idea of HWENO is to evolve both with the solution and its derivative, which allows for using Hermite interpolation in the reconstruction phase, resulting in a more compact stencil at the expense of the additional work. The main difference between this work and the formal one is the procedure to reconstruct the derivative terms. Comparing with the original HWENO schemes of Qiu and Shu, one major advantage of new HWENO schemes is its robust in computation of problem with strong shocks. Extensive numerical experiments are performed to illustrate the capability of the method.