Runge-Kutta Time Discretization Research Articles

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.

High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations

Abstract A new fifth-order, semi-discrete central-upwind scheme with a Lax–Wendroff time discretization procedure for solving Hamilton–Jacobi (HJ) equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. Unlike most of the commonly used high-order upwind schemes, the new scheme is formulated as a Godunov-type method. The new scheme is based on the flux Kurganov, Noelle and Petrova (KNP flux). The spatial discretization is based on a symmetrical weighted essentially non-oscillatory reconstruction of the derivative. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. Various numerical experiments are performed to demonstrate the accuracy and stability properties of the new method. As a result, comparing with other fifth-order schemes for HJ equations, the major advantage of the new scheme is more cost effective for certain problems while the new method exhibits smaller errors without any increase in the complexity of the computations.

A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes

In this paper a third order finite volume weighted essentially non-oscillatory scheme is designed for solving hyperbolic conservation laws on tetrahedral meshes. Comparing with other finite volume WENO schemes designed on tetrahedral meshes, the crucial advantages of such new WENO scheme are its simplicity and compactness with the application of only six unequal size spatial stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, and easy choice of the positive linear weights without considering the topology of the meshes. The original innovation of such scheme is to use a quadratic polynomial defined on a big central spatial stencil for obtaining third order numerical approximation at any points inside the target tetrahedral cell in smooth region and switch to at least one of five linear polynomials defined on small biased/central spatial stencils for sustaining sharp shock transitions and keeping essentially non-oscillatory property simultaneously. By performing such new procedures in spatial reconstructions and adopting a third order TVD Runge–Kutta time discretization method for solving the ordinary differential equation (ODE), the new scheme's memory occupancy is decreased and the computing efficiency is increased. So it is suitable for large scale engineering requirements on tetrahedral meshes. Some numerical results are provided to illustrate the good performance of such scheme.

Runge–Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge–Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $$W^{1,\infty }$$ norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain $$W^{1,\infty }$$ estimates, which are the key to the numerical analysis of these problems.

Approximate Lax–Wendroff discontinuous Galerkin methods for hyperbolic conservation laws

The Lax–Wendroff time discretization is an alternative method to the popular total variation diminishing Runge–Kutta time discretization of discontinuous Galerkin schemes for the numerical solution of hyperbolic conservation laws. The resulting fully discrete schemes are known as LWDG and RKDG methods, respectively. Although LWDG methods are in general more compact and efficient than RKDG methods of comparable order of accuracy, the formulation of LWDG methods involves the successive computation of exact flux derivatives. This procedure allows one to construct schemes of arbitrary formal order of accuracy in space and time. A new approximation procedure avoids the computation of exact flux derivatives. The resulting approximate LWDG schemes, addressed as ALWDG schemes, are easier to implement than their original LWDG versions. In particular, the formulation of the time discretization of the ALWDG approach does not depend on the flux being used. Numerical results for the scalar and system cases in one and two space dimensions indicate that ALWDG methods are more efficient in terms of error reduction per CPU time than LWDG methods of the same order of accuracy. Moreover, increasing the order of accuracy leads to substantial reductions of numerical error and gains in efficiency for solutions that vary smoothly.

A numerical model for three-dimensional shallow water flows with sharp gradients over mobile topography

Abstract This study aims to develop a three-dimensional (3-D) numerical model for shallow water flows over mobile topography, which is capable of simulating morphological evolution under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations (SWEs) using a finite volume method on prismatic cells in σ-coordinates. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. Using a relaxation approach, a hyperbolic system is built for hydrodynamic system, which allows for using a Godunov-type central-upwind method to capture the shocks and approximate the numerical fluxes. Consequently, the 3D-SWEs-Exner model proposed in the present study can stably and accurately solve the dam-break flows over mobile beds. A spatially and temporally second-order “prismatic” central-upwind method is used to approximate the numerical fluxes through cell interfaces. The Exner equation is solved using an upwind method. Using spatially linear reconstruction and explicit two-stage Runge–Kutta time discretization, second order accuracy is achieved in space and time. The proposed model can preserve the well-balanced property due to the special discretization of bed-slope source terms. The proposed model is validated by several tests with experimental measurements, and is compared with the simulated results using reported two-dimensional (2-D) models.

A numerical study of the performance of alternative weighted ENO methods based on various numerical fluxes for conservation law

In this paper, we systematically investigate the performance of the weighted essential non-oscillatory (WENO) methods based on various numerical fluxes for the nonlinear hyperbolic conservation law. Our objective is enhancing the performance for the conservation laws through picking out the suitable numerical fluxes. We focus our attention entirely on the comparison of eight numerical fluxes with the fifth-order accurate finite difference WENO methods and third-order accurate TVD Runge–Kutta time discretization for hyperbolic conservation laws. In addition, we give their implementation based on a new form framework of flux which we used in [9] and was proposed by Shu and Osher in [16]. The detailed numerical study is mainly implemented for the one dimensional system case, including the discussion of the CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations

We construct a local Lax–Friedrichs type positivity-preserving flux for compressible Navier–Stokes equations, which can be easily extended to multiple dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge–Kutta time discretizations satisfy a weak positivity property. With a simple and efficient positivity-preserving limiter, high order explicit Runge–Kutta DG schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy. Numerical tests suggest that the positivity-preserving flux and the positivity-preserving limiter do not induce excessive artificial viscosity, and the high order positivity-preserving DG schemes without other limiters can produce satisfying non-oscillatory solutions when the nonlinear diffusion in compressible Navier–Stokes equations is accurately resolved.

A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation

In the present paper, a high-order finite volume compact scheme is proposed to solve one dimensional Burgers’ equation. The nonlinear advective terms are computed by the fifth-order finite volume weighted upwind compact scheme, in which the nonlinear weighted essentially non-oscillatory weights are coupled with lower order compact stencils. The diffusive terms are discretized by using the finite volume six-order Padé scheme. The strong stability preserving third-order Runge–Kutta time discretizations is used in this work. Numerical results are compared with the exact and some existing numerical solutions to demonstrate the essentially non-oscillatory and high resolution of the proposed method. The numerical results are shown to be more accurate than some numerical results given in the literature.

Stability analysis of the inverse Lax–Wendroff boundary treatment for high order upwind-biased finite difference schemes

In this paper, we consider linear stability issues for one-dimensional hyperbolic conservation laws using a class of conservative high order upwind-biased finite difference schemes, which is a prototype for the weighted essentially non-oscillatory (WENO) schemes, for initial–boundary value problems (IBVP). The inflow boundary is treated by the so-called inverse Lax–Wendroff (ILW) or simplified inverse Lax–Wendroff (SILW) procedure, and the outflow boundary is treated by the classical high order extrapolation. A third order total variation diminishing (TVD) Runge–Kutta time discretization is used in the fully discrete case. Both GKS (Gustafsson, Kreiss and Sundström) and eigenvalue analyses are performed for both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate the stability results predicted by the analysis.

Kronecker product splitting preconditioners for implicit Runge-Kutta discretizations of viscous wave equations

In this paper we study efficient iterative methods for solving the system of linear equations arising from fully implicit Runge-Kutta time discretization of a class of viscous wave equations. In each step of the time integration, a structured system of linear equations is obtained and needs to be solved numerically. A preconditioning strategy based on theKronecker product splitting of the coefficient matrix is applied to solve such linear systems. Some spectral properties of the preconditioned matrix are established and numerical examples are presented to demonstrate the effectiveness of this approach.

Numerical solution of Korteweg-de Vries-Burgers equation by the compact-type CIP method

In this paper, a hybrid compact-CIP scheme is proposed to solve Korteweg-de Vries-Burgers equation. The nonlinear advective terms are computed based on the classical constrained interpolation profile (CIP) method, which is coupled with a high-order compact scheme for third-order derivatives in Korteweg-de Vries-Burgers equation. The strong stability preserving third-order Runge-Kutta time discretizations is adopted in this work. A test case is presented to demonstrate the high-resolution properties of the proposed compact-CIP scheme.

High Order Maximum Principle Preserving Finite Volume Method for Convection Dominated Problems

In this paper, we investigate the application of the maximum principle preserving (MPP) parametrized flux limiters to the high order finite volume scheme with Runge-Kutta time discretization for solving convection dominated problems. Such flux limiter was originally proposed in [Xu, Math. Comp., 2013] and further developed in [Xiong et. al., J. Comp. Phys., 2013] for finite difference WENO schemes with Runge-Kutta time discretization for convection equations. The main idea is to limit the temporal integrated high order numerical flux toward a first order MPP monotone flux. In this paper, we generalize such flux limiter to high order finite volume methods solving convection-dominated problems, which is easy to implement and introduces little computational overhead. More importantly, for the first time in the finite volume setting, we provide a general proof that the proposed flux limiter maintains high order accuracy of the original WENO scheme for linear advection problems without any additional time step restriction. For general nonlinear convection-dominated problems, we prove that the proposed flux limiter introduces up to $O(\Delta x^3+\Delta t^3)$ modification to the high order temporal integrated flux in the original WENO scheme without extra time step constraint. We also numerically investigate the preservation of up to ninth order accuracy of the proposed flux limiter in a general setting. The advantage of the proposed method is demonstrated through various numerical experiments.

Unsteady Convective Heat Transfer of Power-Law Fluid with Variable Fluid Properties in a Concentric Annulus Originating from a Polymer Flooding Process

We study the unsteady convective heat transfer of power-law fluid with variable fluid properties in a concentric annulus with isothermal surface. The problem is originated from the polymer flooding process between a sucker rod and oil well. A new power-law rheological model is proposed, which takes the effects of temperature on fluid viscosity and thermal conductivity into account. Numerical solutions are presented for velocity and temperature fields using the Chebyshev spectral method coupled with the strong stability-preserving Runge–Kutta time discretization. The exponential convergence is verified by accuracy testing between a smooth exact solution of the Partial Differential Equations (PDEs) with source terms and the numerical approximation of manufactured solutions. It is found that heat transfer is enhanced in the variable power-law index model, and a decrease in power-law index of pseudoplastic fluids promotes heat transfer due to the increased Nusselt number. Moreover, the influences of other parameters on convective heat transfer behaviors are discussed in detail.

A weighted Runge–Kutta discontinuous Galerkin method for wavefield modelling

Summary In this paper, we develop a weighted Runge-Kutta discontinuous Galerkin method for the wavefield modeling, which is simply called the WRKDG method. For this method, we first transform the seismic wave equations into a first-order hyperbolic system, and then combine the discontinuous Galerkin spatial discretization with a weighted Runge-Kutta (RK) time discretization. The time discretization is based on a diagonal implicit RK method. We employ an explicit technique to change the implicit RK into an explicit method. We also investigate the stability criteria and numerical dispersion for solving the 2D acoustic equation for this method. Afterwards, we apply the WRKDG method to simulate the wavefields for a 2D homogeneous transversely isotropic (TI) model and a water-filled fracture model. Linear and quadratic interpolations are typically used in this article.

Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations

AbstractThis paper studies a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

Static and dynamic behaviors of belt-drive dynamic systems with a one-way clutch

This paper focuses on the nontrivial equi- librium and the steady-state periodic response of belt- drive system with a one-way clutch and belt flexural rigidity. A nonlinear piecewise discrete-continuous dynamic model is established by modeling the motions of the translating belt spans as transverse vibrations of axially moving viscoelastic beams. The rotations of the pulleys and the accessory are also considered. Further- more, the transverse vibrations and the rotation motions are coupled by nonlinear dynamic tension. The nontriv- ial equilibriums of the belt-drive system are obtained by an iterative scheme via the differential and integral quadrature methods (DQM and IQM). Moreover, the periodic fluctuation of the driving pulley is modeled as the excitation of the belt-drive system. The steady- state periodic responses of the dynamic system are, respectively, studied via the high-order Galerkin trun- cation as well as the DQM and IQM. The time his- tories of the system are numerically calculated based on the 4th Runge-Kutta time discretization method. Furthermore, the frequency-response curves are pre- sented from the numerical solutions. Based on the steady-state periodic response, the resonance areas of the dynamic system are obtained by using the fre- quency sweep. Moreover, the influences of the trun- H. Ding (B)· D.-P. Li

Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces

This paper is concerned with an error analysis for a full discretization of the linear wave equation on a moving surface. The equation is discretized in space by the evolving surface finite element method. Discretization in time is done by Gauβ---Runge---Kutta (GRK) methods, aiming for higher-order accuracy in time and unconditional stability of the fully discrete scheme. The latter is established in the natural time-dependent norms by using the algebraic stability and the coercivity property of the GRK methods together with the properties of the spatial semi-discretization. Under sufficient regularity conditions, optimal-order error estimates for this class of fully discrete methods are shown. Numerical experiments are presented to confirm some of the theoretical results.

Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime

This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.

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.