Stability of Runge-Kutta methods for linear delay differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear scalar delay differential equation . This kind of stability is called stability. We give a characterization of stable Runge-Kutta methods and then we prove that implicit Euler method is stable. Received November 3, 1998 / Revised version received March 23, 1999 / Published online July 12, 2000     

On the asymptotic stability properties of Runge-Kutta methods for delay differential equations

Summary. In this paper asymptotic stability properties of Runge-Kutta (R-K) methods for delay differential equations (DDEs) are considered with respect to the following test equation: where and is a continuous real-valued function. In the last few years, stability properties of R-K methods applied to DDEs have been studied by numerous authors who have considered regions of asymptotic stability for “any positive delay” (and thus independent of the specific value of ). In this work we direct attention at the dependence of stability regions on a fixed delay . In particular, natural Runge-Kutta methods for DDEs are extensively examined. Received April 15, 1996 / Revised version received August 8, 1996     

The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations

Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Received February 10, 1997 / Revised version received January 5, 1998     

Order stars and stability for delay differential equations

Summary. We consider Runge–Kutta methods applied to delay differential equations with real a and b. If the numerical solution tends to zero whenever the exact solution does, the method is called -stable. Using the theory of order stars we characterize high-order symmetric methods with this property. In particular, we prove that all Gauss methods are -stable. Furthermore, we present sufficient conditions and we give evidence that also the Radau methods are -stable. We conclude this article with some comments on the case where a andb are complex numbers. Received June 3, 1998 / Published online: July 7, 1999     

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     

Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Summary. In this paper we present an approach for the numerical solution of delay differential equations where , and , different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1). Received May 4, 1998 / Revised version received January 25, 1999 / Published online November 17, 1999     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

Instability of Runge-Kutta methods when applied to linear systems of delay differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear system of delay differential equations , where . We prove that no Runge-Kutta method preserves asymptotic stability. Received January 24, 2000 / Revised version received July 19, 2000 / Published online June 7, 2001     

On the convergence of Runge–Kutta methods for stiff non linear differential equations

Summary. This paper studies the convergence properties of general Runge–Kutta methods when applied to the numerical solution of a special class of stiff non linear initial value problems. It is proved that under weaker assumptions on the coefficients of a Runge–Kutta method than in the standard theory of B-convergence, it is possible to ensure the convergence of the method for stiff non linear systems belonging to the above mentioned class. Thus, it is shown that some methods which are not algebraically stable, like the Lobatto IIIA or A-stable SIRK methods, are convergent for the class of stiff problems under consideration. Finally, some results on the existence and uniqueness of the Runge–Kutta solution are also presented. Received November 18, 1996 / Revised version received October 6, 1997     

Stability analysis of\theta -methods for neutral functional-differential equations

Summary. This paper deals with the subject of numerical stability for the neutral functional-differential equation It is proved that numerical solutions generated by -methods are convergent if . However, our numerical experiment suggests that they are divergent when is large. In order to obtain convergent numerical solutions when , we use -methods to obtain approximants to some high order derivative of the exact solution, then we use the Taylor expansion with integral remainder to obtain approximants to the exact solution. Since the equation under consideration has unbounded time lags, it is in general difficult to investigate numerically the long time dynamical behaviour of the exact solution due to limited computer (random access) memory. To avoid this problem we transform the equation under consideration into a neutral equation with constant time lags. Using the later equation as a test model, we prove that the linear -method is -stable, i.e., the numerical solution tends to zero for any constant stepsize as long as and , if and only if , and that the one-leg -method is -stable if . We also find out that inappropriate stepsize causes spurious solution in the marginal case where and . Received May 6, 1994     

Rigorous computational shadowing of orbits of ordinary differential equations

Summary. The existence of a true orbit near a numerically computed approximate orbit -- shadowing -- of autonomous system of ordinary differential equations is investigated. A general shadowing theorem for finite time, which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the approximate orbit in terms of computable quantities, is proved. The practical use and the effectiveness of this theorem is demonstrated in the numerical computations of chaotic orbits of the Lorenz equations. Received December 15, 1993     

Numerical solution by LMMs of stiff delay differential systems modelling an immune response

Summary. We consider the application of linear multistep methods (LMMs) for the numerical solution of initial value problem for stiff delay differential equations (DDEs) with several constant delays, which are used in mathematical modelling of immune response. For the approximation of delayed variables the Nordsieck's interpolation technique, providing an interpolation procedure consistent with the underlying linear multistep formula, is used. An analysis of the convergence for a variable-stepsize and structure of the asymptotic expansion of global error for a fixed-stepsize is presented. Some absolute stability characteristics of the method are examined. Implementation details of the code DIFSUB-DDE, being a modification of the Gear's DIFSUB, are given. Finally, an efficiency of the code developed for solution of stiff DDEs over a wide range of tolerances is illustrated on biomedical application model. Received March 23, 1994 / Revised version received March 13, 1995     

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

Summary. In this paper we generalize the class of explicit partitioned Runge-Kutta (PRK) methods for separable Hamiltonian systems to systems with holonomic constraints. For a convenient analysis of such schemes, we first generalize the backward error analysis for systems in to systems on manifolds embedded in . By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suffer from order reduction as well-known for higher-index differential-algebraic equations. However, this order reduction can be avoided by a proper modification of the standard PRK methods. This modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant. We also give a numerical comparison of several second, fourth, and sixth order methods. Received May 5, 1995 / Revised version received February 7, 1996     

Qualocation methods for elliptic boundary integral equations

Summary. The qualocation methods developed in this paper, with spline trial and test spaces, are suitable for classes of boundary integral equations with convolutional principal part, on smooth closed curves in the plane. Some of the methods are suitable for all strongly elliptic equations; that is, for equations in which the even symbol part of the operator dominates. Other methods are suitable when the odd part dominates. Received December 27, 1996 / Revised version received April 14, 1997     

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     

Preserving monotonicity in the numerical solution of Riccati differential equations

Summary. Solutions of symmetric Riccati differential equations (RDEs for short) are in the usual applications positive semidefinite matrices. Moreover, in the class of semidefinite matrices, solutions of different RDEs are also monotone, with respect to properly ordered data. Positivity and monotonicity are essential properties of RDEs. In Dieci and Eirola (1994), we showed that, generally, a direct discretization of the RDE cannot maintain positivity, and be of order greater than one. To get higher order, and to maintain positivity, we are thus forced to look into indirect solution procedures. Here, we consider the problem of how to maintain monotonicity in the numerical solutions of RDEs. Naturally, to obtain order greater than one, we are again forced to look into indirect solution procedures. Still, the restrictions imposed by monotonicity are more stringent that those of positivity, and not all of the successful indirect solution procedures of Dieci and Eirola (1994) maintain monotonicity. We prove that by using symplectic Runge-Kutta (RK) schemes with positive weights (e.g., Gauss schemes) on the underlying Hamiltonian matrix, we eventually maintain monotonicity in the computed solutions of RDEs. Received May 2, 1995     

Stability of Runge-Kutta methods for the generalized pantograph equation

Summary. This paper deals with stability properties of Runge-Kutta (RK) methods applied to a non-autonomous delay differential equation (DDE) with a constant delay which is obtained from the so-called generalized pantograph equation, an autonomous DDE with a variable delay by a change of the independent variable. It is shown that in the case where the RK matrix is regular stability properties of the RK method for the DDE are derived from those for a difference equation, which are examined by similar techniques to those in the case of autonomous DDEs with a constant delay. As a result, it is shown that some RK methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation. Stability of algebraically stable natural RK methods is also considered. Received May 5, 1998 / Revised version received November 17, 1998 / Published online September 24, 1999     

Corrected collocation methods for periodic pseudodifferential equations

Summary. We propose an approximation method for periodic pseudodifferential equations, which yields higher convergence rates in Sobolev spaces with negative order than the collocation method. The main idea consists in correcting the usual collocation solution in a certain way by the solution of a small Galerkin system for the same equation. Both trigonometric and spline approximation methods are considered. In most of the cases our convergence result even improves that of the qualocation method. Received January 3, 1994 / Revised version received August 17, 1994     

Spline qualocation methods for variable-coefficient elliptic equations on curves

Summary. Here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type. The analysis exploits localization in the form of frozen coefficients, pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators, a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders. Numerical examples show the predicted high orders of convergence. Received January 29, 1998 / Published online: June 29, 1999     

Positive definiteness in the numerical solution of Riccati differential equations

Summary. In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix differential equations. In many cases -- typical in applications -- the solutions are positive definite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The first class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show first that using a direct algorithm -- a one-step scheme or a strictly stable multistep scheme (explicit or implicit) -- limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite. Then we show two ways to obtain positive definite higher order approximations by using indirect algorithms. The first is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization. Received April 21, 1993