The evaluation of numerical techniques for solution of stiff ordinary differential equations arising from chemical kinetic problems

Ordinary differential equations with widely scattered eigenvalues (stiff O.D.E.'s) occur often in the studies of reaction network problems. Five numerical methods, including two methods based on Backward differentiation formulas, a modified Runge-Kutta-Fehlberg method, a method based on PECE Adams formulas, and an improved semi-implicit Euler method are evaluated by comparing their performance when applied to test systems. The test systems represent different combinations of linearity and nonlinearity, small and large dimension, real and complex eigenvalues, and slightly stiff and very stiff problems. The relative merits and dificiencies of the methods are discussed.     

Split-step Adams–Moulton Milstein methods for systems of stiff stochastic differential equations

In this paper we discuss new split-step methods for solving systems of Itô stochastic differential equations (SDEs). The methods are based on a L-stable (strongly stable) second-order split Adams–Moulton Formula for stiff ordinary differential equations in collusion with Milstein methods for use on SDEs which are stiff in both the deterministic and stochastic components. The L-stability property is particularly useful when the drift components are stiff and contain widely varying decay constants. For SDEs wherein the diffusion is especially stiff, we consider balanced and modified balanced split-step methods which posses larger regions of mean-square stability. Strong order convergence one is established and stability regions are displayed. The methods are tested on problems with one and two noise channels. Numerical results show the effectiveness of the methods in the pathwise approximation of stiff SDEs when compared to some existing split-step methods.     

Convergence Results of One-Leg and Linear Multistep Methods for Multiply Stiff Singular Perturbation Problems

One-leg methods and linear multistep methods are two class of important numerical methods applied to stiff initial value problems of ordinary differential equations. The purpose of this paper is to present some convergence results of A-stable one-leg and linear multistep methods for one-parameter multiply stiff singular perturbation problems and their corresponding reduced problems which are a class of stiff differential-algebraic equations. Received April 14, 2000; revised June 30, 2000     

MOL solvers for hyperbolic PDEs with source terms

A method-of-lines solution algorithm for reacting flow problems modelled by hyperbolic partial differential equations (PDEs) with stiff source terms is presented. Monotonicity preserving advection schemes are combined with space/time error balancing and a Gauss–Seidel iteration to provide an efficient solver. Numerical experiments on two challenging examples are presented to illustrate the performance of the method.     

Solving stiff differential equations for simulation

Computer simulation of dynamic systems very often leads to the solution of a set of stiff ordinary differential equations. The solution of this set of equations involves the eigenvalues of its Jacobian matrix. The greater the spread in eigenvalues, the more time consuming the solutions become when existing numerical methods are employed. Extremely stiff differential equations can become a very serious problem for some systems, rendering accurate numerical solutions completely uneconomic. In this paper, we propose new techniques for solving extremely stiff systems of differential equations. These algorithms are based on a class of implicit Runge-Kutta procedure with complete error estimate. The new techniques are applied to solving mathematical models of the relaxation problem behind blast waves.     

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.     

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.     

Third-order composite Runge–Kutta method for stiff problems

The derivation of a composite method for solving stiff ordinary differential equations is discussed. Combination of the harmonic and arithmetic means of the Runge–Kutta formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. The numerical results and the A-stability of this new formula are examined.     

Numerical methods of boundary layer type for stiff systems of differential equations

Stiff systems of ordinary differential equations are difficult to deal with numerically. There is an equivalence between a subclass of stiff systems and differential equations subjected to singular perturbations. We use the characterization of the solution of this class of equations in terms of boundary layers as a means of generating numerical procedures for solving the stiff equations. The numerical procedures have the desirable feature of improving with increasing stiffness.     

Numerical integration of semidiscrete evolution systems

Methods and algorithms for integrating initial value systems are examined. Of particular interest is efficient and accurate numerical integration of systems of ordinary differential equations that arise on semidiscrete spatial differencing or finite element projection for evolution problems characterized by partial differential equations. Integration schemes for general systems are described. Stiff and oscillatory systems are considered and these motivate selection of specific types of algorithms for certain problem classes. For example, we show that Runge-Kutta methods with extended regions of stability are particularly efficient for moderately stiff dissipative systems derived from parabolic transport equations. The theoretical developments of an earlier paper [1] determine bounds on stiffness and stability and may be used to examine the stiff dissipative or oscillatory nature of the system qualitatively. Order control and stepsize adjustment in variable-order, variable-step algorithms are compared for several integrators applied to stiff and nonstiff initial-value systems arising from representative parabolic evolution problems.     

Stability and convergence analysis of implicit–explicit one-leg methods for stiff delay differential equations

The purpose of this paper is devoted to studying the implicit–explicit (IMEX) one-leg methods for stiff delay differential equations (DDEs) which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff DDEs are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods.     

Study of general Taylor-like explicit methods in solving stiff ordinary differential equations

In this paper, we present a further study of Taylor-like explicit methods in solving stiff ordinary differential equations. We derive the general form for Taylor-like explicit methods in solving stiff differential equations. We also analyse the order of convergence and stability property for the general form. Moreover, we give its corresponding vector form via introducing a new definition of vector product and quotient in another article. The convergence and stability of the vector form are considered as well.     

An efficient algorithm for solving stiff ordinary differential equations

The Brayton-Gustavson-Hatchel (BGH) method for solving stiff ordinary differential equations belongs to the group of backward difference formulas methods. Basic details of the BGH method are presented. A new implementation with original modifications is described. Special attention is paid to the reduction of operation count and improvement of error control. Two examples including mildly stiff and stiff equation systems prove spectacular superiority of the BGH method with respect to the classic Gear method. The software presented in this paper is scalable and has been ported without any problems from the PC/DOS platform to two UNIX environments.     

A type insensitive code for delay differential equations basing on adaptive and explicit Runge-Kutta interpolation methods

For the numerical solution of initial value problems for delay differential equations with constant delay a partitioned Runge-Kutta interpolation method is studied which integrates the whole system either as a stiff or as a nonstiff one in subintervals. This algorithm is based on an adaptive Runge-Kutta interpolation method for stiff delay equations and on an explicit Runge-Kutta interpolation method for nonstiff delay equations. The retarded argument is approximated by appropriate Lagrange or Hermite interpolation. The algorithm takes advantage of the knowledge of the first points of jump discontinuities. An automatic stiffness detection and a stepsize control are presented. Finally, numerical tests and comparisons with other methods are made on a great number of problems including real-life problems.     

Implementing Radau IIA Methods for Stiff Delay Differential Equations

This article discusses the numerical solution of a general class of delay differential equations, including stiff problems, differential-algebraic delay equations, and neutral problems. The delays can be state dependent, and they are allowed to become small and vanish during the integration. Difficulties encountered in the implementation of implicit Runge–Kutta methods are explained, and it is shown how they can be overcome. The performance of the resulting code – RADAR5 – is illustrated on several examples, and it is compared to existing programs. Received October 12, 2000     

基于IMEX的织物动态仿真的近似解法

织物在空间运动的刚性特征始终是困扰织物动态仿真的难题．显式方法简单灵活，易于实现，但受稳定因素影响，无法实现具有刚性特征的织物动态模拟；隐式方法稳定性好，却忽略了非线性因素，而且计算复杂，直接影响到仿真的最终结果和实际效率．针对这一问题，提出了基于隐式一显式的近似解法，该方案从系统受力形变的非线性特征出发，将质点受力分为线性和非线性两部分，线性部分采用隐式解法，非线性部分利用显式解法，线性方程组的求解则运用近似解法．实验结果表明，该方法兼具两种方法的优点，既保留了隐式方法的稳定性，又充分利用了显式方法的简易性处理非线性特征，从而从真正意义上解决织物仿真中的刚性问题．     

无线传感器网络安全路由算法的研究与设计

无线传感器网络安全路由问题是亟需解决和具有重要意义的课题,是无线传感器网络安全的一个重要研究方向.本文针对GEAR路由算法的防御弱点,设计了一种新的安全路由算法R-GEAR,通过在GEAR路由算法上引入适用于传感器网络的安全引导方案及信誉评测机制来获得较好的安全性能.对R-GEAR进行安全性分析,并利用NS2仿真工具进行仿真实验.结果表明,在存在攻击节点的情况下,与GEAR路由算法相比,R-GEAR安全路由算法具有更好的包传输率、包丢失率.     

Improving nonlinear electronic circuit simulation

The resolution of systems of stiff differential equations is required in the transient analysis of a large electronic network simulation. Resultant stability problems and the methods used in solving first order stiff nonlinear differential equations are reviewed. An improved algorithm is presented using BDF formulas given by Brayton et al. IEEE Vol 60 (1972) pp 98–108 and has been implemented in the IMAG electronic circuit simulation program. Reducing computer time has been achieved by controlling the number of Newton iterations, the number of integration steps, and the number of Jacobian matrix evaluations without producing additional errors or instability phenomena. Experimental results are shown.     

Using Algorithms of Decomposition for Computation of Linear Stochastic Models

The mathematical apparatus of decomposition is used to solve the problem of analysis and computation of stiff stochastic systems of differential equations. A theorem substantiating the adequacy of a solution obtained is formulated and an algorithm of computation of stiff stochastic systems by the method of depression of equations is given.     

Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs

The methods for the dynamical simulation of multi-body systems in real-time applications have to guarantee that the time integration of the equations of motion is always successfully completed within an a priori fixed sampling time interval, typically in the range of 1.0–10.0 ms. Model structure, model complexity and numerical solution methods have to be adapted to the needs of real-time simulation. Standard solvers for stiff and for constrained mechanical systems are implicit and cannot be used straightforwardly in real-time applications because of their iterative strategies to solve the nonlinear corrector equations and because of adaptive strategies for stepsize and order selection. As an alternative, we consider in the present paper noniterative fixed stepsize time integration methods for stiff ordinary differential equations (ODEs) resulting from tree-structured multi-body system models and for differential algebraic equations (DAEs) that result from multi-body system models with loop-closing constraints.