Applicability of the ECM Performance Model to Explicit ODE Methods on Current Multi-core Processors

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge---Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge---Kutta stability is also briefly discussed

Stochastic differential equation models play an increasingly prominent role in a wide variety of application areas, particularly those characterized by complexity and random behavior. Examples of such areas include biology, epidemiology, mechanics, economics, and finance. This issue's Education section contains a paper, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations," by Desmond Higham, that makes this important material accessible to a wide range of readers. Higham's paper is written from the perspective that the best way to gain insight quickly into the topics of stochastic differential equations (SDEs) is to see examples of numerical methods in use. The author feels that it is possible to explain and understand how simple numerical methods in this area work, without requiring the reader to be familiar with SDE theory. Indeed, the paper takes the approach that simulations of the dynamic behavior of SDEs and experimentation with numerical examples can help develop an intuitive understanding of SDEs that lays the groundwork for subsequent theoretical treatments. Thus Higham's paper expects only that the reader's background includes Euler's method for deterministic ordinary differential equations and an intuitive understanding of random variables. It provides a beautifully written treatment that is aimed at upper-division undergraduates and beginning graduate students. The topics covered in Higham's paper include investigation of stochastic integration, linear stability, and strong and weak convergence. The paper is built around ten complete downloadable MATLAB programs that are used as examples and contain ample comment lines. The programs are linked to relevant discussion points in the body of the paper. This paper is suitable as a module in a numerical methods course, such as a course on numerical methods for ordinary differential equations. It could also find application in courses in the areas where SDEs arise, such as finance, epidemiology, or general mathematical modeling. And it can serve excellently as the starting point for a self-study project in the area of SDEs for advanced undergraduates or beginning graduate students. Finally, many professionals in the SIAM community will find this article useful as an introduction to an area of numerical computation that has grown in prominence since many of us were students!

We consider an enzyme-substrate reaction-diffusion problem. Unsteady and steady state models are recalled. For the unsteady state case, the model is in the form of a second order partial differential equation. We solve the unsteady state model using the explicit numerical finite difference method, which is forward difference in time and centered difference in space. For the general steady state case, the model is in the form of a second order ordinary differential equation. We solve the general steady state model using the explicit first order Euler's numerical method. For the particular steady state case of the unsaturated catalytic kinetics, we derive the exact analytical solution using the characteristic method of ordinary differential equations. For the particular steady state case of the saturated catalytic kinetics, we derive the exact analytical solution using the direct-integration method. The obtained exact analytical solutions are identical with the existing exact analytical solutions derived using the variational iteration method. With the aid of computer, the enzyme-substrate reaction-diffusion problem can be solved and simulated successfully for both unsteady and steady state cases.

Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of stability regions defined by using scalar test equations. The analysis is closely related to the stability analysis of the standard linear multistep methods for delay differential equations. A new second-order IMEX method which has approximately the same stability region as that of the IMEX Euler method, the simplest IMEX method of order 1, is proposed. Some numerical results are also presented which show superiority of the new method.

Continuous numerical methods for ODEs with defect control

Comparative Stochastic Convergence Analysis of Seven Recursive Parameter Estimation Methods

Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd.

In this research, the main objective is to perform the comparative analysis of numerical methods (Explicit Euler, RungeKutta 4 and LocallyExact) for the resolution of differential equations. To fulfill the purpose of this study, the system ofdifferential equations of the Lotka-Volterra model was used and the mathematical software Wolfram Mathematica wasused. To perform the comparison of the numerical methods the Lotka-Volterra model was solved using the NdSolvecommand of Mathematica, this result was compared with the Methods Explicit Euler, Runge Kutta 4 and LocallyExact. Theresults obtained from the phase diagrams and the point table of the interactions indicate that the Runge Kutta 4 methodhas greater precision, followed by the LocallyExact method. The explicit Euler method draws considerably away from theresult of NDSolve.

This research undertakes a comprehensive investigation of the optical soliton solutions of the Focusing Non-linear Schr¨odinger Equation (NLSE), a fundamental model describing the propagation of optical solitons in nonlinear media. To achieve this, we employ two versatile and efficient methods: the Ricatti-Bernoulli Sub Ordinary Differential Equation (RBSODE) method and the Bernoulli Sub Ordinary Differential Equation (BSODE) method. These methods enable us to derive a wide range of optical soliton solutions.We examine two distinct nonlinearities: the Kerr law nonlinearity and the quadratic-cubic nonlinearity. These nonlinearities are crucial in determining the behavior of optical solitons in various nonlinear optical media. Our analysis reveals that the derived soliton solutions exhibit distinct characteristics, such as amplitude, width, and velocity, which are influenced by the type of nonlinearity and the parameters of the NLSE.Furthermore, we utilize the multiplier approach to obtain the conservation laws of the NLSE. These conservation laws provide valuable insights into the underlying dynamics of the optical solitons and have significant implications for the design and optimization of nonlinear optical system.

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di- stability) in problem discretizations. However, computational cate- gories are not precise, and qualitative descriptors such as of moderate size cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive New- ton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate bound- ary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood.

Many classes of methods for numerical problems can be characterized by parameters. For example, Adams methods for integrating ordinary differential equations can be characterized by the step size and order. In many cases it is known that a method is stable for fixed values of those parameters, whereas many programs vary those parameters in order to reduce computation time. This paper examines the effects on stability of changes to two kinds of parameters—“discrete” and “continuous” — corresponding to the parameters of order and step size in the numerical solution of ordinary differential equations. Conditions are given that are sufficient to ensure that these changes do not affect stability. These results are applied to two classes of automatic methods for ordinary differential equations—one based on unequal interval quadrature formulas, the other based on interpolation and equal interval quadrature formulas.

An exponentially-fitted method for singularly-perturbed ordinary differential equations with turning points and parabolic problems

A class of Galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed. For polynomials of degree k , these methods lead to a family of one-step schemes generating approximations up to order 2 k + 2 2k + 2 for the solution of an ordinary differential equation.

II.A - Exact Methods for ODEs

This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consistency, stability, and convergence of integration methods are introduced and studied. This allows for a fruitful application of the results obtained in Chapter 12. Subsequently, one-step methods are considered in detail and their important properties are shown. Moreover, Runge–Kutta methods are thoroughly investigated. Finally, we deal with linear multistep methods and study the asymptotic behavior of integration methods and of stiff differential equations. In particular, we deal with Dahlquist’s root condition and derive the first Dahlquist barrier and the second Dahlquist barrier.