Some Modified Runge-Kutta Methods for the Numerical Solution of Initial-Value Problems with Oscillating Solutions

Two new modified Runge-Kutta methods with minimal phase-lag are developed for the numerical solution of initial-value problems with oscillating solutions which can be analyzed to a system of first order ordinary differential equations. These methods are based on the well known Runge-Kutta RK5(4)7FEq1 method of Higham and Hall (1990) of order five. Also, based on the property of the phase-lag a new error control procedure is introduced. Numerical and theoretical results show that this new approach is more efficient compared with the well known Runge-Kutta Dormand-Prince RK5(4)7S method [see Dormand and Prince (1980)] and the well known Runge-Kutta RK5(4)7FEq1 method of Higham and Hall (1990).     

Parallel-Iterated RK-Type PC Methods With Continuous Output Formulas * This work was partly supported by N.R.P.F.S.

This paper investigates parallel predictor-corrector iteration schemes (PC iteration schemes) based on collocation Runge-Kutta corrector methods (RK corrector methods) with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The resulting parallel-iterated RK-type PC methods are also provided with continuous output formulas. The continuous numerical approximations are used for predicting the stage values in the PC iteration processes. In this way, we obtain parallel PC methods with continuous output formulas and high-accurate predictions. Applications of the resulting parallel PC methods to a few widely-used test problems reveal that these new parallel PC methods are much more efficient when compared with the parallel and sequential explicit RK methods from the literature.     

An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

A Fifth Order Runge-Kutta RK(5, 5) Method With Error Control

In this paper a new Runge-Kutta RK(5, 5) method is introduced. The theory and analysis of its properties are investigated and compared with the more well known RKF(4, 5) and RK(4, 5) - Merson methods.     

On high order strong stability preserving runge-kutta and multi step time discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.     

基于龙格库塔法的对抗攻击方法

深度神经网络在许多领域中取得了显著的成果, 但相关研究结果表明, 深度神经网络很容易受到对抗样本的影响. 基于梯度的攻击是一种流行的对抗攻击, 引起了人们的广泛关注. 研究基于梯度的对抗攻击与常微分方程数值解法之间的关系, 并提出一种新的基于常微分方程数值解法-龙格库塔法的对抗攻击方法. 根据龙格库塔法中的预测思想, 首先在原始样本中添加扰动构建预测样本, 然后将损失函数对于原始输入样本和预测样本的梯度信息进行线性组合, 以确定生成对抗样本中需要添加的扰动. 不同于已有的方法, 所提出的方法借助于龙格库塔法中的预测思想来获取未来的梯度信息(即损失函数对于预测样本的梯度), 并将其用于确定所要添加的对抗扰动. 该对抗攻击具有良好的可扩展性, 可以非常容易地集成到现有的所有基于梯度的攻击方法. 大量的实验结果表明, 相比于现有的先进方法, 所提出的方法可以达到更高的攻击成功率和更好的迁移性.     

Dispersion and Dissipation Error in High-Order Runge-Kutta Discontinuous Galerkin Discretisations of the Maxwell Equations

Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.     

The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods

Splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A three-stage second-order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard s-stage explicit RK schemes of order s, s = 2, 3. It has advantages in the low accuracy range, and this range is interesting for an application in splitting methods. Numerical results are presented.     

带输入延时的连续时间系统的离散化

含多项式插值的Runge-Kutta方法应用于对带输入延时的连续时间系统的离散化中. 与传统的离散化方法相比, 本文提出的方法是有效且精度高阶的. 此方法的精度与Runge-Kutta法及插值多项式的精度紧密相关. 本文讨论了离散化方法的近似精度阶及最大可达的精度阶. 除此之外, 也分析了方法的输入状态稳定性. 为保证相应离散系统的稳定性, 可通过考察RK法的绝对稳定域来选择采样时间. 特别当RK法是A-稳定时, 可以不受稳定性的约束选择采样时间. 最后提供了一个数值例子来证明方法的优越性.     

显式和对角隐式Rung-Kutta方法求解中立型泛函微分方程的非线性稳定性

本文致力于研究巴拿赫空间中非线性中立型泛函微分方程显式和对角隐式Rung-Kutta方法的稳定性．获得了一些显式和对角隐式Rung-Kutta方法求解非线性中立型泛函微分方程的数值稳定性和条件收缩性结果,数值试验验证了这些结果．     

Starting algorithms for a class of RK methods for index-2 DAEs

When semiexplicit differential-algebraic equations are solved with implicit Runge-Kutta methods (RK), the computational effort is dominated by the cost of solving the nonlinear systems, and therefore it is important to have good starting values to begin the iterations. For semiexplicit index-2 DAEs, starting algorithms without additional cost for RK methods with regular matrix coefficient were studied in a previous paper. However, the regularity condition on the matrix coefficient excludes some interesting methods like Lobatto IIIa and ESDIRK methods. In this paper, we study starting algorithms, without additional computational cost, for a class of Runge-Kutta methods in the case of index-2 DAEs.     

Second-order stabilized explicit Runge-Kutta methods for stiff problems

Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10⁵). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.     

On iterated Crank-Nicolson methods for hyperbolic and parabolic equations

Different versions of the iterated Crank-Nicolson method are considered and their relation to the explicit Runge-Kutta methods is studied. Stability and accuracy properties of these methods are compared for both hyperbolic and parabolic equations. It is shown that Runge-Kutta methods offer more accurate and stable options even for a few evaluations of the right-hand sides. When applied to nonlinear equations, the iterated Crank-Nicolson methods have an accuracy barrier, which does not appear for Runge-Kutta methods.     

A fourth-order symplectic exponentially fitted integrator

A numerical method for ordinary differential equations is called symplectic if, when applied to Hamiltonian problems, it preserves the symplectic structure in phase space, thus reproducing the main qualitative property of solutions of Hamiltonian systems. In a previous paper [G. Vanden Berghe, M. Van Daele, H. Van de Vyver, Exponential fitted Runge-Kutta methods of collocation type: fixed or variable knot points?, J. Comput. Appl. Math. 159 (2003) 217-239] some exponentially fitted RK methods of collocation type are proposed. In particular, three different versions of fourth-order exponentially fitted Gauss methods are described. It is well known that classical Gauss methods are symplectic. In contrast, the exponentially fitted versions given in [G. Vanden Berghe, M. Van Daele, H. Van de Vyver, Exponential fitted Runge-Kutta methods of collocation type: fixed or variable knot points?, J. Comput. Appl. Math. 159 (2003) 217-239] do not share this property. This paper deals with the construction of a fourth-order symplectic exponentially fitted modified Gauss method. The RK method is modified in the sense that two free parameters are added to the Buthcher tableau in order to retain symplecticity.     

One-step implicit methods or solving delay differential equations

Two one-step implicit methods—the second order Trapezium method and the fourth order implicit Runge-Kutta method for solving the delay differential equations (DDE) are developed. The significance of implicit methods lie in their 4-stability for ordinary differential equations. Different techniques are used to approximate the delay term. We also discuss the local truncation error estimate. Numerical examples are solved to show the effectiveness of the methods so developed.     

Construction and implementation of general linear methods for ordinary differential equations: A review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge-Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge-Kutta stability is also briefly discussed.     

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

In this paper nonlinear monotonicity and boundedness properties are analyzed for linear multistep methods. We focus on methods which satisfy a weaker boundedness condition than strict monotonicity for arbitrary starting values. In this way, many linear multistep methods of practical interest are included in the theory. Moreover, it will be shown that for such methods monotonicity can still be valid with suitable Runge-Kutta starting procedures. Restrictions on the stepsizes are derived that are not only sufficient but also necessary for these boundedness and monotonicity properties.     

计算微分代数系统的实时仿真算法

§１．引言 对于由微分代数方程所表示的动力系统的数值算法,针对微分代数方程的一些特殊形式已经构造了一些有效算法如文献[1]-［４］．这些数值算法大部分都是基于常微分方程的一些隐式公式如隐式Ｒｕｎｇｅ－Ｋｕｔｔａ方法,向后微分公式（ＢＤＦ）等,因此这些算法都是非实时仿真算法．如果我们直接用求解常微分方程的显式公式如显式 Ｒｕｎｇｅ－Ｋｕｔｔａ方法,显式线性多步法等,虽然满足了实时仿真算法的一些特点,但是这些数值公式对微分代数方程的求解不甚理想．由于一个实时仿真算法具有实时性、周期性、可靠性等特性要求,因…     

Numerical study of one-step explicit-implicit methods of L-equivalent stiffly accurate two-stage Runge-Kutta schemes

The paper describes one-step methods for numerical integration of the Cauchy problem for systems of ordinary differential equations free from iterations and coinciding on linear problems (autonomous and non-autonomous) with stiffly accurate implicit two-stage Runge-Kutta (RK) schemes. The numerical study of their accuracy is performed on stiff tests, i.e., the autonomous Kaps system and non-autonomous Protero-Robinson problem.     

On modified Runge-Kutta trees and methods

Modified Runge-Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge-Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested.