Order conditions for numerical methods for partitioned ordinary differential equations

Summary Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of P-series is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods fory=f(y,y), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].     

Mean-square convergence of numerical methods for random ordinary differential equations

Numerical Algorithms - The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the...     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

Hybrid BDF methods for the numerical solutions of ordinary differential equations

In this article, we have presented the details of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solutions of ordinary differential equations (ODEs). In these hybrid BDF, one additional stage point (or off-step point) has been used in the first derivative of the solution to improve the absolute stability regions. Stability domains of our presented methods have been obtained showing that all these new methods, we say HBDF, of order p, p = 2,4,..., 12, are A(α)-stable whereas they have wide stability regions comparing with those of some known methods like BDF, extended BDF (EBDF), modified EBDF (MEBDF), adaptive EBDF (A-EBDF), and second derivtive Enright methods. Numerical results are also given for five test problems.     

One-step methods for ordinary differential equations

Summary It is proved that any consistent one-step method for solving the initial value problem for a first-order ordinary differential equation is convergent; no stability condition is required. An application is made to a similarly stated result, allowing part of the hypothesis in that case to be dropped.     

Chebyshevian multistep methods for ordinary differential equations

Summary In this paper some theory of linear multistep methods fory ^(r) (x)=f(x,y) is extended to include smooth, stepsize-dependent coefficients. Treated in particular is the case where exact integration of a given set of functions is desired.Work on this paper was supported in part by U.S. Army Research Office (Durham) Grant DA-ARO(D)-31-124-G1050 and National Science Foundation Grant GP-23655 with The University of Texas at Austin.     

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.     

Projection methods and approximations for ordinary differential equations

A formulation of a differential equation as projection and fixed point problem allows approximations using general piecewise functions. We prove existence and uniqueness of the approximate solution, convergence in the L₂ norm and nodal superconvergence. These results generalize those obtained earlier by Hulme for continuous piecewise polynomials and by Del four-Dubeau for discontinuous piecewise polynomials. A duality relationship for the two types of approximations is also given. This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (Grant OGPLN-336) and by the “Ministère de l’Education du Québec” (FCAR Grant-ER-0725).     

Error bounds for numerical solutions of ordinary differential equations

Summary An a posteriori error bound, for an approximate solution of a system of ordinary differential equations, is derived as the solution of a Riccati equation. The coefficients of the Riccati equation depend on an eigenvalue of a matrix related to a Jacobian matrix, on a Lipschitz constant for the Jacobian matrix, and on the approximation defect. An upper bound is computable as the formal solution of a sequence of Riccati equations with constant coefficients. This upper bound may sometimes be used to control step length in a numerical method.     

Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

     

Two-step almost collocation methods for ordinary differential equations

A new class of two-step Runge-Kutta methods for the numerical solution of ordinary differential equations is proposed. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. Local error estimation for these methods is also discussed.     

The numerical solution of unstable ordinary differential equations

Invariant imbedding, or the Riccati transformation, has been used to solve unstable ordinary differential equations for a few years. This paper compares the above method with parallel or multiple shooting and a method using Chebyshev series. Parallel shooting gives a solution as accurate as that obtained using the Riccati transformation, in a comparable time.     

Unified approach to numerical methods. III. Finite differences and ordinary differential equations

This is the third of a sequence of articles devoted to the presentation of a direct method of analysis recently developed by one of the authors. The approach is quite general, since it is applicable to any linear operator, symmetric or nonsymmetric, regardless of its type. In particular, the theory includes steady-state and time-dependent problems. In this article the method is applied to ordinary differential equations, which constitute a very convenient illustration of its application because the analysis can be carried out in an exhaustive manner. General algorithms which yield the exact values of the solution or its derivatives are obtained in this manner.     

Nonlinear methods in solving ordinary differential equations

Some one step methods, based on nonpolynomial approximations, for solving ordinary differential equations are derived, and numerically tested. A comparison is made with existing methods.     

Two-step numerical methods for parabolic differential equations

A family of two-stepA-stable methods of maximal order for the numerical solution of ordinary differential systems is developed. If these methods are applied to the stiff, large systems which originate from linear parabolic differential equations they yield a large, sparse set of linear algebraic equations of special form. This set is considerably easier to solve than the algebraic equations which are obtained when using diagonal Obrechkoff methods, which are one-step,A-stable and of maximal order     

Numerical methods for ordinary differential equations on matrix manifolds

In recent years differential systems whose solutions evolve on manifolds of matrices have acquired a certain relevance in numerical analysis. A classical example of such a differential system is the well-known Toda flow. This paper is a partial survey of numerical methods recently proposed for approximating the solutions of ordinary differential systems evolving on matrix manifolds. In particular, some results recently obtained by the author jointly with his co-workers will be presented. We will discuss numerical techniques for isospectral and isodynamical flows where the eigenvalues of the solutions are preserved during the evolution and numerical methods for ODEs on the orthogonal group or evolving on a more general quadratic group, like the symplectic or Lorentz group. We mention some results for systems evolving on the Stiefel manifold and also review results for the numerical solution of ODEs evolving on the general linear group of matrices.     

Diagonally implicit general linear methods for ordinary differential equations

We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, p=q orq+1,s=r+1=q, p=q orq+1, ands=r=q, p=q+1. Examples of such methods are constructed with stability function matching theA-acceptable generalized Padé approximations to the exponential function.The work of Z. Jackiewicz was partially supported by the National Science Foundation under grant NSF DMS-9208048.     

Improved linear multi-step methods for stochastic ordinary differential equations

We consider linear multi-step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise. We present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations. In previous work, we considered Maruyama-type schemes, where only the increments of the driving Wiener process are used to discretize the diffusion part. Here, we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical-stochastic integrals. We show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile. Simulation results illustrate the theoretical findings.