The RKGL method for the numerical solution of initial-value problems

We introduce the RKGL method for the numerical solution of initial-value problems of the form y^′=f(x,y)$y^{\prime }=f(x,y)$, y(a)=α$y(a)=\alpha$. The method is a straightforward modification of a classical explicit Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f(x,y)$f(x,y)$ needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ah^r+1+Bh^2m${\mathit{Ah}}^{r+1}+{\mathit{Bh}}^{2m}$, where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.     

The numerical analysis on a Volterra equation with asymptotically periodic solution

We consider order one operational quadrature methods on a certain integro-differential equation of Volterra type on (0,∞), with piecewise linear convolution kernels. The forms of discretization solution are patterned after a continuous one of Hannsgen (1979) [2]. An l¹ remainder stability and an error bound are derived.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Stability analysis of numerical methods for delay differential equations

Summary This paper deals with the stability analysis of step-by-step methods for the numerical solution of delay differential equations. We focus on the behaviour of such methods when they are applied to the linear testproblemU(t)=U(t)+U(t–) with >0 and , complex. A general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods.     

A note on the Gautschi-type method for oscillatory second-order differential equations

The Gautschi-type method has been proposed by Hochbruck and Lubich for oscillatory second-order differential equations. They conjecture that this method allows for a uniform error bound independent of the size of the system. The conjecture is proved in this note.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

A mountain pass method for the numerical solution of semilinear wave equations

Summary It is well-known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals. In the situation that we studied, there is usually an obvious solution obtained as a solution of linear problem. We formulate a dual variational problem in such a way that the obvious solution is a local minimum. We then find additional non-obvious solutions via a numerical mountain pass algorithm, based on the theorems of Ambrosetti, Rabinowitz and Ekeland. Numerical results are presented.Research supported in part by grant DMS-9208636 from the National Science FoundationResearch supported in part by grant DMS-9102632 from the National Science Foundation     

A periodic solution of a differential equation with state-dependent delay

We study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future. The main result is existence of a periodic solution in case the equilibrium is linearly unstable. The proof employs the ejective fixed point principle on a compact convex set K₀⊂C([−h,0],R) of Lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite-dimensional submanifold of the space C¹([−h,0],R).     

A Gautschi-type method for oscillatory second-order differential equations

Summary. We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of García-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation. Received January 21, 1998 / Published online: June 29, 1999     

A numerical method to boundary value problems for second order delay differential equations

Summary In the first part of this paper we are dealing with theoretical statements and conditions which lead to existence and uniqueness of the solution of a nonlinear boundary value problem with delay. Next we apply this method successfully to a numerical example. The computations have been carried out at the computer Siemens 4004. The data obtained are presented in two tables.     

A delay differential equation solver based on a continuous Runge–Kutta method with defect control

We have recently developed a generic approach for solving neutral delay differential equations based on the use of a continuous Runge–Kutta formula with defect control and investigated its convergence properties. In this paper, we describe a method, DDVERK, which implements this approach and justify the strategies and heuristics that have been adopted. In particular we show how the assumptions related to error control, stepsize control, and discontinuity detection (required for convergence) can be efficiently realized for a particular sixth-order numerical method. Summaries of extensive testing are also reported. This revised version was published online in August 2006 with corrections to the Cover Date.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

API stepsize control for the numerical solution of ordinary differential equations

A control-theoretic approach is used to design a new automatic stepsize control algorithm for the numerical integration of ODE's. The new control algorithm is more robust at little extra expense. Its improved performance is particularly evident when the stepsize is limited by numerical stability. Comparative numerical tests are presented.     

Analytical solution of the linear fractional differential equation by Adomian decomposition method

In this paper, we consider the n-term linear fractional-order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by Adomian decomposition method. With the equivalent transmutation, we show that the solution by Adomian decomposition method is the same as the solution by the Green's function. Finally, we illustrate our result with some examples.     

A qualocation method for Burgers’ equation

In this paper, a qualocation method for the one-dimensional Burgers’ equation is proposed. A semidiscrete scheme along with optimal error estimates is discussed. Results of a numerical experiment performed support the theoretical results.     

Exponential mean square stability of numerical methods for systems of stochastic differential equations

This paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split-step theta method for stochastic systems. First, we consider linear autonomous systems. Under a sufficient and necessary condition for exponential mean square stability of the exact solution, it is proved that the two classes of theta methods with θ≥0.5$\theta \ge 0.5$ are exponentially mean square stable for all positive step sizes and the methods with θ<0.5$\theta <0.5$ are stable for some small step sizes. Then, we study the stability of the methods for nonlinear non-autonomous systems. Under a coupled condition on the drift and diffusion coefficients, it is proved that the split-step theta method with θ>0.5$\theta >0.5$ still unconditionally preserves the exponential mean square stability of the underlying systems, but the stochastic theta method does not have this property. Finally, we consider stochastic differential equations with jumps. Some similar results are derived.