Ordinary differential equation convergence

The basic notions of the theory of difference schemes are the error of approximation, stability, convergence, and accuracy of difference scheme. A more detailed exposition of these eoncepts will appear in Chapter 2. They are illustrated by considering a number of difference schemes for ordinary differential equations. In the same chapter we also outline the approach to the general formulations without regard to the particular form of the difference operator. 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

The second type includes, for example, the problem investigated by Gud-erley (cited in [4]) of convergence of a cylindrical shock wave to a line, or of a spherical wave to a point. In this case, just finding the exponent requires integration of ordinary differential equations. The exponent is found from the condition that the integral curve passes through a singular point without this it is impossible to satisfy the boundary conditions. 

The set of four ordinary differential equations (7.64) to (7.67) for the dynamical system are quite sensitive numerically. Extreme care should be exercised in order to obtain reliable results. We advise our students to experiment with the standard IVP integrators ode... in MATLAB as we have done previously in the book. In particular, the stiff integrator odel5s should be tried if ode45 turns out to converge too slowly and the system is thus found to be stiff by numerical experimentation. 

In this subsection, some commonly used numerical schemes that involve difference equations to solve ordinary differential equations are presented along with their stability characteristics. Simple examples to illustrate the effects of step size on the convergence of numerical methods are shown. A simple discretization of the first-order linear differential equation... 

In this section we discuss the issues of convergence and accuracy in numerical integration methods for solving ordinary differential equations. 

The Verlet method is a numerical method that respects certain conservation principles associated to the continuous time ordinary differential equations, i.e. it is a geometric integrator. Maintaining these conservation properties is essential in molecular simulation as they play a key role in maintaining the physical environment. As a prelude to a more general discussion of this topic, we demonstrate here that it is possible to derive the Verlet method from the variational principle. This is not the case for every convergent numerical method. The Verlet method is thus a special type of numerical method that provides a discrete model for classical mechanics. 

The numerical-analytic method for the investigation of periodic solutions of nonlinear systems of ordinary differential equations has been proposed by one of the authors. This method enables us to construct periodic solutions for these systems in the form of uniformly convergent sequences of periodic functions. In the monograph, we justify the applicability of the numerical-analytic method to the investigation of periodic solutions for various classes of nonlinear systems with heredity. We also clarify the influence of a lag on the existence of these solutions. 

After the spatial discretization for System A and B a system of 512 ordinary differential equations is obtained. Hence N = 512 for the first two systems. In Fig. 1, left graph, the error Si ( = F (a i) — Xi) ) versus the CPU time in seconds is given for the dynamic simulation of System A and B. We see that System A converges very slowly (769 cycle simulations, or iterations of F) to a periodic state, whereas System B converges very fast (31 iterations of F). 

The discretization of the model equations results for System D in a system of 120 ordinary differential equations. Hence N = 120 for System D. System D is unique among the test systems in that the convergence behaviour of the different methods depends on the initial condition. All the methods are started from two different initial states. The... 

It turns out that steady states (more commonly referred to af fixed points or equilibria) of 1-D maps have stability properties like the steady states of ordinary differential equations. We can ask whether, if we choose x infinitesimally close to, but not exactly at, the fixed point, the sequence will approach the fixed point or move away from it. If subsequent numbers in the sequence converge toward the fixed point, then it is stable. If not, then it is unstable. If there is more than one stable fixed point, then each of these points will have its own basin of attraction, just as we found for ordinary differential equations with multiple stable states. Without attempting to derive it (it is not difficult), we simply present the result that a fixed point x of a 1-D map/(x ) is stable if the derivative df jdx at the fixed point X lies between —1 and 1, that is. 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

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