Runge-Kutta methods for quadratic ordinary differential equations

Many systems of ordinary differential equations are quadratic: the derivative can be expressed as a quadratic function of the dependent variable. We demonstrate that this feature can be exploited in the numerical solution by Runge-Kutta methods, since the quadratic structure serves to decrease the number of order conditions. We discuss issues related to construction design and implementation and present a number of new methods of Runge-Kutta and Runge-Kutta-Nyström type that display superior behaviour when applied to quadratic ordinary differential equations.     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …     

Classification of Local Conservation Laws of Maxwell's Equations

A complete and explicit classification of all independent local conservation laws of Maxwell's equations in four dimensional Minkowski space is given. Besides the elementary linear conservation laws, and the well-known quadratic conservation laws associated to the conserved stress-energy and zilch tensors, there are also chiral quadratic conservation laws which are associated to a new conserved tensor. The chiral conservation laws possess odd parity under the electric–magnetic duality transformation of Maxwell's equations, in contrast to the even parity of the stress-energy and zilch conservation laws. The main result of the classification establishes that every local conservation law of Maxwell's equations is equivalent to a linear combination of the elementary conservation laws, the stress-energy and zilch conservation laws, the chiral conservation laws, and their higher order extensions obtained by replacing the electromagnetic field tensor by its repeated Lie derivatives with respect to the conformal Killing vectors on Minkowski space. The classification is based on spinorial methods and provides a direct, unified characterization of the conservation laws in terms of Killing spinors.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Runge-Kutta theory for Volterra integrodifferential equations

Summary The present paper develops the theory of general Runge-Kutta methods for Volterra integrodifferential equations. The local order is characterized in terms of the coefficients of the method. We investigate the global convergence of mixed and extended Runge-Kutta methods and give results on asymptotic error expansions. In a further section we construct examples of methods up to order 4.     

Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter . The coefficients in the expansion of the global error in powers of are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.     

A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any one-step collocation method is equivalent to one of such methods. In this paper, we consider a linear constant-coefficient system of DDEs with a constant delay, and discuss the application of natural Runge-Kutta methods to the system. We show that anA-stable method preserves the asymptotic stability property of the analytical solutions of the system.     

Multirate Runge–Kutta schemes for advection equations

Explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena. The wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step. Multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps, resulting in a moderate to substantial reduction of the computational cost, depending on the scenario to simulate [S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput. 41 (1983) 321–336; H. Tang, G. Warnecke, A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and pace grids, SIAM J. Sci. Comput. 26 (4) (2005) 1415–1431; E. Constantinescu, A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, SIAM J. Sci. Comput. 33 (3) (2007) 239–278]. In air pollution modeling the advection part is usually integrated explicitly in time, where the time step is constrained by a locally varying Courant–Friedrichs–Lewy (CFL) number. Multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction. Therefore it is of major interest to apply multirate schemes to the advection equation. We introduce a generic recursive multirate Runge–Kutta scheme that can be easily adapted to an arbitrary number of refinement levels. It preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit Runge–Kutta methods as base method.     

基于坏单元指示子的p自适应RKDG算法

Runge-Kutta间断Galerkin(RKDG)方法是求解双曲守恒律方程的主流方法之一.很多双曲守恒律问题的规模特别巨大,数值计算求解时需要耗费大量的时间和计算机存储空间.为此我们设计了一个基于坏单元指示子的p自适应RKDG算法,它不仅能够保持原RKDG方法在光滑区域的高阶逼近,而且能够有效降低存储空间.同时,这也为后续进一步开展hp自适应RKDG算法的研究奠定了基础.     

Runge-Kutta methods for jump-diffusion differential equations

In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

Runge-kutta schemes for Hamiltonian systems

We study the application of Runge-Kutta schemes to Hamiltonian systems of ordinary differential equations. We investigate which schemes possess the canonical property of the Hamiltonian flow. We also consider the issue of exact conservation in the time-discretization of the continuous invariants of motion.     

Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1

Summary For the numerical solution of non-stiff semi-explicit differentialalgebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods.     

Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2

Usually the straightforward generalization of explicit Runge-Kutta methods for ordinary differential equations to half-explicit methods for differential-algebraic systems of index 2 results in methods of orderq≤2. The construction of higher order methods is simplified substantially by a slight modification of the method combined with an improved strategy for the computation of the algebraic solution components. We give order conditions up to orderq=5 and study the convergence of these methods. Based on the fifth order method of Dormand and Prince the fifth order half-explicit Runge-Kutta method HEDOP5 is constructed that requires the solution of 6 systems of nonlinear equations per step of integration.     

On quadratic invariants and symplectic structure

We show that the theorems of Sanz-Serna and Eirola and Sanz-Serna concerning the symplecticity of Runge-Kutta and Linear Multistep methods, respectively, follow from the fact that these methods preserve quadratic integral invariants and are closed under differentiation and restriction to closed subsystems.     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Convergence of general linear methods on differential-algebraic systems of index 3

Many numerical methods used to solve ordinary differential equations or differential-algebraic equations can be written as general linear methods. The purpose of this paper is to extend the known convergence results for Runge-Kutta and linear multistep methods to a large class of new promising numerical schemes. The theoretical results are illustrated by some numerical experiments.     

On first-order conservation laws for systems of hydrodynamic-type equations

First-order conservation laws quadratic in derivatives are considered for systems of hydrodynamic-type equations. Defining relationships for the densities of such conservation laws are derived in a form that is invariant with respect to pointwise changes of the variables. Examples of nondiagonalizable systems admitting quadratic conservation laws are given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 109–128, July, 1996.     

Convergence of a class of runge-kutta methods for differential-algebraic systems of index 2

This paper deals with convergence results for a special class of Runge-Kutta (RK) methods as applied to differential-algebraic equations (DAE's) of index 2 in Hessenberg form. The considered methods are stiffly accurate, with a singular RK matrix whose first row vanishes, but which possesses a nonsingular submatrix. Under certain hypotheses, global superconvergence for the differential components is shown, so that a conjecture related to the Lobatto IIIA schemes is proved. Extensions of the presented results to projected RK methods are discussed. Some numerical examples in line with the theoretical results are included.     

Error of rosenbrock methods for stiff problems studied via differential algebraic equations

This paper studies Rosenbrock methods when they are applied to stiff differential equations containing a small stiffness parameter. The basic ideas and techniques are the same as those developed for Runge-Kutta methods in an earlier paper of the authors. The results obtained here are essentially those obtained for Diagonally Implicit Runge-Kutta methods.