Analysis of the dynamics of local error control via a piecewise continuous residual

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way. Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228. Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant DMS-9201727.     

Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples.     

A Note on the Numerical Solution of Quasi-linear Parabolic Partial Differential Equations with Error Estimates

A method is considered for the numerical solution of quasi-linearpartial differential equations. The partial differential equationis reduced to a set of ordinary differential equations usinga Chebyshev series expansion. The exact solution of this setof ordinary differential equations is shown to be the solutionof a perturbed form of the original equation. This enables errorestimates to be found for linear and mildly non-linear problems.     

Error propagation in numerical approximations near relative equilibria

We study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of Hamiltonian differential equations with symmetries. First it is shown that taking an initial perturbation of a relative equilibrium, the corresponding solution is related, in a first approximation, to another relative equilibrium, with the parameters perturbed from the original. Then, this is used to prove that, for stable relative equilibria, error growth with respect to the perturbed solution is in general quadratic, but only linear for schemes that preserve the invariant quantities of the problem. In this sense, the conclusion is similar to the one obtained when integrating unperturbed relative equilibria. Numerical experiments illustrate the results.     

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     

Runge–Kutta methods and Banach algebras

The equations defining both the exact and the computed solution to an initial value problem are related to a single functional equation, which can be regarded as prototypical. The functional equation can be solved in terms of a formal Taylor series, which can also be generated using an iteration process. This leads to the formal Taylor expansions of the solution and approximate solutions to initial value problems. The usual formulation, using rooted trees, can be modified to allow for linear combinations of trees, and this gives an insight into the nature of order conditions for explicit Runge–Kutta methods. A short derivation of the family of fourth order methods with four stages is given.     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

Short memory principle and a predictor–corrector approach for fractional differential equations

Fractional differential equations are increasingly used to model problems in acoustics and thermal systems, rheology and modelling of materials and mechanical systems, signal processing and systems identification, control and robotics, and other areas of application. This paper further analyses the underlying structure of fractional differential equations. From a new point of view, we apprehend the short memory principle of fractional calculus and farther apply a Adams-type predictor–corrector approach for the numerical solution of fractional differential equation. And the detailed error analysis is presented. Combining the short memory principle and the predictor–corrector approach, we gain a good numerical approximation of the true solution of fractional differential equation at reasonable computational cost. A numerical example is provided and compared with the exact analytical solution for illustrating the effectiveness of the short memory principle.     

Linearly implicit extrapolation methods for differential-algebraic systems

Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.     

A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations

This study deals with the singularly perturbed initial value problem for a quasilinear first-order delay differential equation. A numerical method is generated on a grid that is constructed adaptively from a knowledge of the exact solution, which involves appropriate piecewise-uniform mesh on each time subinterval. An error analysis shows that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines

Summary.   The collocation tension spline is considered as a numerical solution of a singularly perturbed two-point boundary value problem: . The collocation points are chosen as a generalization of the classical Gaussian points. Unlike the traditional approach, we employ the B-spline representation in the analysis. This leads to global quadratic convergence of the method for small perturbation parameters, and, for large values, the order of convergence is four. Received October 4, 1996 / Revised version received September 23, 1999 / Published online October 16, 2000     

Runge–Kutta convolution quadrature methods for well-posed equations with memory

Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided.     

A singularly perturbed problem of photon transport in a time‐dependent region

The paper is devoted to the numerical study of a singularly perturbed transport linear integro‐differential equation, in a time‐dependent domain with slab geometry. After a brief summary on existence and uniqueness results for such a model, we test the error between the exact solution and its quasi‐static approximation, which satisfies a simpler equation. Copyright © 2002 John Wiley & Sons, Ltd.     

A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems

A system of coupled singularly perturbed initial value problems with two small parameters is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solution of the system has boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh a hybrid finite difference scheme is proved to be almost second-order accurate, uniformly in both small parameters. Numerical results supporting the theory are presented.     

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.     

Numerical method for verifying the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed Gelfand equation

A numerical verification method to confirm the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed Gelfand equation is presented. Using certain systems of equations corresponding to a double turning point, we derive a sufficient condition for its existence whose satisfaction can be verified computationally. We describe verification procedures and give a numerical example as a demonstration.     

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     

An improved approximate Newton method for implicit Runge-Kutta formulas

Implicit Runge-Kutta (IRK) methods (such as the s-stage Radau IIA method with s=3,5, or 7) for solving stiff ordinary differential equation systems have excellent stability properties and high solution accuracy orders, but their high computing costs in solving their nonlinear stage equations have seriously limited their applications to large scale problems. To reduce such a cost, several approximate Newton algorithms were developed, including a commonly used one called the simplified Newton method. In this paper, a new approximate Jacobian matrix and two new test rules for controlling the updating of approximate Jacobian matrices are proposed, yielding an improved approximate Newton method. Theoretical and numerical analysis show that the improved approximate Newton method can significantly improve the convergence and performance of the simplified Newton method.