Linearly implicit quantization-based integration methods for stiff ordinary differential equations

In this paper, new integration methods for stiff ordinary differential equations (ODEs) are developed. Following the idea of quantization-based integration (QBI), i.e., replacing the time discretization by state quantization, the proposed algorithms generalize the idea of linearly implicit algorithms. Also, the implementation of the new algorithms in a DEVS simulation tool is discussed. The efficiency of these new methods is verified by comparing their performance in the simulation of two benchmark problems with that of other numerical stiff ODE solvers. In particular, the advantages of these new algorithms for the simulation of electronic circuits are demonstrated.     

Multistep-multistage-multiderivative methods for ordinary differential equations

This paper studies a general method for the numerical integration of ordinary differential equations. The method, defined in part 1, contains many known processes as special case, such as multistep methods, Runge-Kutta methods (“multistage”), Taylor, series (“multiderivative”) and their extensions (section 2). After a short section on trees and pairs of trees we derive formulas for the conditions to be satisfied by the free parameters in order to equalize the numerical approximation with the solution up to a certain order. Next we extend the reuslts of Kastlunger [6]. The proof given here is shorter than the original one. Finally we discuss formulas, with the help of which the conditions for the parameters can be reduced considerably and give numerical examples.     

Parallel methods for ordinary differential equations

Remarkably few methods have been proposed for the parallel integration of ordinary differential equations (ODEs). In part this is because the problems do not have much natural parallelism (unless they are virtually uncoupled systems of equations, in which case the method is obvious). In part it is because the subproblems arising in the solution of ODEs (for example, the solution of linear equations) are the ones that have provided the challenges for parallelism. This paper surveys some of the methods that have been proposed, and suggests some additional methods that are suitable for special cases, such as linear problems. It then looks at the possible application of large-scale parallelism, particularly across the method. If efficiency is of no concern (that is, if there is an arbitrary number of proceessors) there are some ways in which the solution of stiff equations can be done more rapidly; in fact, a speed up from a parallel time of 0(N ²) to 0(logN) forN equations might be possible if communication time is ignored. This is obtained by trying to perform as much as possible of the matrix arithmetic associated with the solution of the linear equations at each step in advance of that step and in parallel with the integration of earlier steps.     

General linear methods for ordinary differential equations

General linear methods were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.     

An iterative integration scheme for solving second-order ordinary differential equations

This paper presents an iterative solution method for the numerical integration of second-order ordinary differential equations using a simple program for microcomputers (PC). The method of integration proposed is based on the geometrical considerations in the phase plane. The numerical results are compared to those obtained by the fourth-order Runge-Kutta method and to the closed form solutions when possible. Tests show good accuracy and, in some cases, computer time saving with respect to the Runge-Kutta's method for th same accuracy. The method of integration in the phase plane seems very good for treating every kind of nonlinear second-order differential equation whatever the degree of nonlinearity.     

Singly-implicit Runge-Kutta methods for retarded and ordinary differential equations

A continuous singly-implicit Runge-Kutta method is implemented for stiff retarded differential equations. The choice of this implicit Runge-Kutta method is based on stability investigations of wide classes of interpolationintegration schemes. The numerical results show the effectiveness of these methods for both stiff ordinary and retarded differential equations.     

Error estimation in the integration of ordinary differential equations

Linear initial value problems, particularly involving first order differential equations, can be transformed into systems of higher order and treated as boundary value problems. Finite difference analogues considered for obtaining approximate solutions of these boundary value problems are proved to be fourth order convergent processes, by deriving considerable sharper bounds for the discretization error. Numerical examples are given to demonstrate the usefulness of our error bounds.     

Efficient automatic integration of ordinary differential equations with discontinuities

This paper describes a method of automatically detecting and accurately locating discontinuities which occur in many applications of ordinary differential equations. The integration formula is a Runge-Kutta so chosen that accurate values between integration points can be found by Hermite interpolation.The efficiency of the method arises from two sources: (i) the classification of discontinuities into two types, known as time and state events; and (ii) the location of state events using Hermite interpolation.     

Rational Runge-Kutta methods for solving systems of ordinary differential equations

Some nonlinear methods for solving single ordinary differiential equations are generalized to solve systems of equations. To perform this, a new vector product, compatible with the Samelson inverse of a vector, is defined. Conditions for a given order are derived.     

Uniqueness for ordinary differential equations

Various criteria are known for assuring uniqueness of the solution of a system ofn ordinary differential equations,x = f(t, x), with initial conditionx(t ₀) = x₀. Most of these involve some sort of relaxed Lipschitz condition onf(t, x), with respect tox, valid on an open setD R ¹⁺ⁿ which contains the point (t ₀, x₀). The present paper generalizes (and unifies) a number of known uniqueness criteria to cover cases when (t ₀, x₀) lies on the boundary ofD. Research partially supported by NSF Grant GP-37838.     

A class of hybrid collocation methods for third-order ordinary differential equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x _n+j , i = 0(1)k and at an off grid point x = x _n+u , where k is the step number of the method and u is an arbitrary rational number in (x _n , x _n+k ). A predictor of order 2k ? 1 is also proposed to cater for y _n+k in the main method. Taylor series expansion is employed for the calculation of y _n+1, y _n+2, y _n+u and their higher derivatives. Evaluation of the resulting method at x = x _n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.     

Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

New extrapolation methods for initial value problems in ordinary differential equations

In Ikhile (1998), a family of variable order inverse nonlinear rational methods for which the inverse-Euler is a special case has been developed. This present consideration examines the extrapolation of this class of methods. In fact, we show that in the limit of increasing order, the extrapolation of the generalised class of methods remain highly stable. In particular, the L-stability of the inverse Euler remains a preserved property after polynomial extrapolation. The accuracy of the numerical results from extrapolation based on these methods appear quite overwhelming when compared to results from DIFEX2 of Fatunla (1986,1988), DIFEX1 of Deuflhard (1983) and GBS extrapolation method of Gragg (1965) and Bulirsch and Stoer (1966).     

New stability theorems concerning one-step numerical methods for ordinary differential equations

A measure of the stability properties of numerical integration methods for ordinary differential equations is provided by their stability region, which is that region in the complex (Δtλ) plane for which a given method is stable when applied to the differential equation

$\text{d}\text{y}\text{d}\text{t=λy}$

with a time-step Δt.Free parameters which exist in numerical integration algorithms may be used to maximize, in some sense, the size of the stability region, rather than increasing the order of accuracy, as is usually done. We derive new results which set theoretical limits to this maximization process for one step, explicit methods.Specifically, if K is the number of function evaluation invoked, then:(i) we prove (Theorem 1) that if ? is the radius of the largest circle, tangent to the imaginary axis at the origin of the complex plane that is contained in the stability region S, then ? cannot exceed K.(ii) we also prove (Theorem 2) that the imaginary stability boundary S_I (or maximum stable value of ∣Δtλ∣ with λ imaginary) cannot exceed (K ? 1).While Theorem 1 is to our knowledge new, a limited form of theorem 2 (K odd only) had been established in v.d. Houwen (1977). That the maximum imaginary boundary S_I = (K ? 1) is attainable had been shown (constructively) for K odd. We show that this maximum is also reached for K = 2 and K = 4, and correct in the process an erroneous result in the above reference.     

Some methods of integrating ordinary differential equations on a multiprocessor computer

Cybernetics and Systems Analysis - The methods discussed herein for use on single- and multiprocessor computers have approximately the same error characteristics. The stability boundaries of the...     

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

Solving ordinary differential equations for simulation

Runge-Kutta formulas are given which are suited to the tasks arising in simulation. They are methods permitting interpolation which use overlap into the succeeding step to reduce the cost of a step and its error estimate.