Contractivity of Runge-Kutta methods

In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value problems, but also conditional contractivity for initial value problems where the right hand side function satisfies a circle condition. Our results are relevant for arbitrary norms, in particular for the maximum norm.For contractive methods, we also focus on the question whether there exists a unique solution to the algebraic equations in each step. Further we show that contractive methods have a limited order of accuracy. Various optimal methods are presented, mainly of explicit type. We provide a numerical illustration to our theoretical results by applying the method of lines to a parabolic and a hyperbolic partial differential equation.Research supported by the Netherlands Organization for Scientific Research (N.W.O.) and the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.)     

Contractivity of locally one-dimensional splitting methods

Summary The aim of this paper is to study contractivity properties of two locally one-dimensional splitting methods for non-linear, multi-space dimensional parabolic partial differential equations. The term contractivity means that perturbations shall not propagate in the course of the time integration process. By relating the locally one-dimensional methods with contractive integration formulas for ordinary differential systems it can be shown that the splitting methods define contractive numerical solutions for a large class of non-linear parabolic problems without restrictions on the size of the time step.     

Contractive methods for stiff differential equations part I

An integration method for ordinary differential equations is said to be contractive if all numerical solutions of the test equationx=x generated by that method are not only bounded (as required for stability) but non-increasing. We develop a theory of contractivity for methods applied to stiff and non-stiff, linear and nonlinear problems. This theory leads to the design of a collection of specific contractive Adams-type methods of different orders of accuracy which are optimal with respect to certain measures of accuracy and/or contractivity. Theoretical and numerical results indicate that some of these novel methods are more efficient for solving problems with a lack of smoothness than are the familiar backward differentiation methods. This lack of smoothness may be either inherent in the problem itself, or due to the use of strongly varying integration steps. In solving smooth problems, the efficiency of the low-order contractive methods we propose is approximately the same as that of the corresponding backward differentiation methods.This work was done during the first author's stay at the IBM Thomas J. Watson Research Center under his appointment as Senior Researcher of the Academy of Finland. It was sponsored in addition by the IBM Corporation and by the AirForce Office of Scientific Research (AFSC), United States Air Force, under contracts No. F44620-75-C-0058 and F49620-77-C-0088. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.     

Note on Vector Translation with Respect to Thompson’s Metric

It is known that vector translations are contractive with respect to Thompson’s part metric. Here, we give a simple proof, based on a representation of Thompson’s metric through positive functionals. Moreover, we use contractivity of translations to prove a fixed point result for mappings that are Lipschitz continuous with respect to Thompson’s metric with Lipschitz constant r>1. The case r = 1 for order preserving or order reversing mappings has been recently studied by Lawson and Lim. We apply our result to a nonlinear boundary value problem.     

Contractivity of θ-method for semi-discrete systems

We consider nonlinear semi-discrete problems that derive by reaction diffusion systems of partial differential equations, when finite difference methods or Faedo Galerkin methods are used for spatial discretization. The aim of this article is to give sufficient conditions for the contractivity of the θ-method, in a norm generated by a positive diagonal matrix G. We show that the numerical contractivity property is obtained if some matrices, constructed by means of the Jacobian matrix of nonlinear term, are M-matrices. © 1996 John Wiley & Sons, Inc.     

A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions

We study the comparison principle for degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L¹ contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kru?kov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic-hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L¹ contractive property.     

A note on contractivity in the numerical solution of initial value problems

This paper concerns the stability analysis of numerical methods for approximating the solutions to (stiff) initial value problems. Our analysis includes the case of (nonlinear) systems of differential equations that are essentially more general than the classical test equationU=U, with a complex constant.We explore the relation between two stability concepts, viz. the concepts of contractivity and weak contractivity.General Runge-Kutta methods, one-stage Rosenbrock methods and a notable rational Runge-Kutta method are analysed in some detail.     

求解多延迟微分方程的Runge-Kutta方法的收缩性

该文涉及多延迟微分方程MDDEs系统的理论解与数值解的收缩性．为此,一些新的稳定性概念诸如：BN_f^(m)-稳定性及GRN_m-稳定性稳定性被引入．该探讨得出：Ｒｕｎｇｅ Ｋｕｔｔａ（ＲＫ）方法及相应的连续插值的BN^(m)-稳定性导致求解MDDEs的方法的收缩性（GRN_m-稳定性）．     

Comparing the contractivity properties of semi-implicit methods

Contractivity is a desirable property of numerical integration methods for stiff systems of ordinary differential equations. In this paper, numerical parameters are used to allow a direct and quantitative comparison of the contractivity properties of various methods for non-linear stiff problems. Results are provided for popular Rosenbrock methods and some more recently developed semi-implicit methods.     

Completely contractive representations for some doubly generated antisymmetric operator algebras

Contractive weak star continuous representations of the Fourier binest algebra (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of by semicrossed product algebras and on the complete contractivity of contractive representations of such algebras. The latter result is obtained by two applications of the Sz.-Nagy-Foias lifting theorem. In the presence of an approximate identity of compact operators it is shown that an automorphism of a general weakly closed operator algebra is necessarily continuous for the weak star topology and leaves invariant the subalgebra of compact operators. This fact and the main result are used to show that isometric automorphisms of the Fourier binest algebra are unitarily implemented.

     

A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations

In this review, we present the recent work of the author in comparison with various related results obtained by other authors in literature. We first recall the stability, contractivity and asymptotic stability results of the true solution to nonlinear stiff Volterra functional differential equations (VFDEs), then a series of stability, contractivity, asymptotic stability and B-convergence results of Runge-Kutta methods for VFDEs is presented in detail. This work provides a unified theoretical foundation for the theoretical and numerical analysis of nonlinear stiff problems in delay differential equations (DDEs), integro-differential equations (IDEs), delayintegro-differential equations (DIDEs) and VFDEs of other type which appear in practice.      

A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14

A family of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite–Birkhoff–Taylor methods of order p =?5,6,…,14, that we denote by CPHBT_RK4(d,s,p), with nonnegative coefficients are constructed by casting s-stage Runge–Kutta methods of order 4 with Taylor methods of order d. The constructed CPHBT_RK4 methods are implemented using efficient variable step control and are compared to other well-known methods on a variety of initial value problems. A selected method: CP 6-stages 9-derivative HBT method of order 12, denoted by CPHBTRK412, has larger region of absolute stability than Dormand–Prince DP(8,7)13M and Taylor method T(12) of order 12. It is superior to DP(8,7)13M and T(12) methods on the basis the number of steps, CPU time, and maximum global error on several problems often used to test higher-order ODE solvers. Also, we show that the contractivity preserving property of CPHBTRK412is very efficient in suppressing the effect of the propagation of discretization errors and the new method compares positively with explicit 17 stages Runge-Kutta-Nyström pair of order 12 by Sharp et al. on a long-term integration of a standard N-body problem. The selected CPHBTRK412is listed in the Appendix.     

Contractivity preserving explicit linear multistep methods

Summary We investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on the general test-equation , whereA is a square matrix of arbitrary orders1. The contractivity is analysed with respect to arbitrary norms in thes-dimensional space (which are not necessarily generated by an inner product). For given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize.This research has been supported by the Netherlands organisation for scientific research (NWO)     

The growth of a C 0-semigroup characterised by its cogenerator

We characterise contractivity, boundedness and polynomial growth for a C ₀-semigroup in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomée and show that polynomial growth of a semigroup implies polynomial growth of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial growth of the cogenerators implies polynomial growth of the semigroup. In addition, we show by simple examples in (), , that our results on the characterization of contractivity are sharp. These examples also show that the famous Foiaş-Sz.-Nagy theorem on cogenerators of contractive C ₀-semigroups on Hilbert spaces fails in () for .      

Index 2 DAEs in Plasticity Theory

The equations of rate‐independent elastoplasticity form a differential‐algebraic equation (DAE) with discontinuities. For the numerical solution, implicit Runge‐Kutta methods are applied and combined with the return mapping strategy of computational plasticity. It turns out that the convergence order depends crucially on the switching point detection. Further, it is shown that algebraically stable Runge‐Kutta methods preserve the contractivity of the elastoplastic flow.     

On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups

We use functional calculus methods to investigate qualitative properties of C₀-semigroups that are preserved by time-discretization methods. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. Preservation of contractivity (or norm-bound) of the semigroup is investigated in the Banach space setting. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. Since the H-P functional calculus is used throughout the paper, we present an elementary introduction to it based on the Riemann-Stieltjes integral.     

Conditional contractivity of Runge–Kutta methods for nonlinear differential equations with many variable delays

This paper deals with conditional contractivity properties of Runge–Kutta (RK) methods with variable step-size applied to nonlinear differential equations with many variable delays (MDDEs). The concepts of CRN_m(ω, H)- and BN_f(μ, ?)-stability are introduced. It is shown that the numerical solution produced by a BN_f(μ, ?)-stable Runge–Kutta method with an appropriate interpolation is contractive. In particular, these results are also novel for nonlinear differential equations with many constant delays or single variable delay. To obtain BN_f(μ, ?)-stable methods, (k, l)-algebraically stable Runge–Kutta methods are also investigated.     

一族多步二阶导数方法的收缩性

１．引言 １９７８年 Ｎｅｖａｎｌｉｎｎａ和 Ｌｉｎｉｇｅｒ［１,２］研究了常微分方程初值问题的单支方法和线性多步法的收缩性,就基于线性模型方程的收缩性建立了比较完整的理论．他们指出,收缩方法比绝对稳定方法能更好地给出间断问题的数值解,因而研究数值方法的收缩性具有重要理论和实践意义． １９７４年 Ｅｎｒｉｇｈｔ［３］构造了 ｋ步 ｋ ＋ ２阶二阶导数方法由于它是Ａｄｍａｓ型的且只含一个二阶导数项,因而方法在原点附近具有较理想的稳定性和稳定程度（参见［７］）,同时在 ∞处是极端稳定的．赵双锁和董国雄 ［４］…     

On the Contractivity and Asymptotic Stability of Systems of Delay Differential Equations of Neutral Type

In this paper we investigate both the contractivity and the asymptotic stability of the solutions of linear systems of delay differential equations of neutral type (NDDEs) of the form y(t) = Ly(t) + M(t)y(t – (t)) + N(t)y(t – (t)). Asymptotic stability properties of numerical methods applied to NDDEs have been recently studied by numerous authors. In particular, most of the obtained results refer to the constant coefficient version of the previous system and are based on algebraic analysis of the associated characteristic polynomials. In this work, instead, we play on the contractivity properties of the solutions and determine sufficient conditions for the asymptotic stability of the zero solution by considering a suitable reformulation of the given system. Furthermore, a class of numerical methods preserving the above-mentioned stability properties is also presented.