Solving initial-value problems

Due to the preceding remarks, we can limit our considerations to the case of one first-order I VP n = 1) in one function or state variable y R — R and one independent variable x, starting at x = a with the initial condition y(a) ya R. We want to solve the given IVP for the unknown function y(x) and the given right hand side F x,y). In other words, we seek the values of y(x) for all values of x in the interval [a, b. Expressed in mathematical notation, we thus seek to solve 

Here F and ya are known. The function y(x) K — R is unknown on the interval [a, b], except for its initial value y a) = ya, called the initial condition. 

The fundamental theorem of calculus allows us to rewrite the left hand side as y(x +1) — y(xi). Thus for each i we obtain 

Numerical ODE solvers use this equation to find approximations yi+ for the value y(xi+i) of y at xi+i from F, yi, and possibly other data. Sophisticated ODE solvers use several previously computed approximate values yi-2,. .. of y(xl i), y(x 2), 

There are more refined one-step integrators than Euler s method. Multi-step integration methods use more than just one previously computed y and F value and thereby they can better account for y s curvature and higher derivatives. Thus they follow the actual solution curve much closer. But in turn, they need more supporting F data computations. 

In order to give an introduction to the most important concepts used in different methods, such as accuracy, stability, and efficiency, we will start with the exphcit Euler method, which is the simplest numerical method to use when solving ODEs. At this point, we should stress that the accuracy of this method is low, and that it is only conditionally stable. For this reason, it is not used in practice to solve any engineering problems. However, it serves well to illustrate a method for solving ODEs numerically, and for introducing the concepts of accuracy and stability. Later in this chapter more advanced numerical methods will be presented methods that have higher accuracy and better stability properties, and that are implemented in modem software products. 

The error made in the Euler method in a single step, s, can be estimated by using the Taylor expansion 

A simple problem illustrates how the error is related to step size. Consider the following problem  

The midpoint method requires two function evaluations in the step, 

The fourth-order Runge-Kutta method is sometimes used because the method is accurate, stable, and easy to program. However, there are much better alternatives, as we will soon discover. This method uses four function evaluations per step. 

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The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

To find u, it is necessary to use some ancillary equations. As usual in solving initial value problems, we assume that all variables are known at the reactor inlet so that (Ac)i UinPin will be known. Equation (3.2) can be used to calculate m at a downstream location if p is known. An equation of state will give p but requires knowledge of state variables such as composition, pressure, and temperature. To find these, we will need still more equations, but a closed set can eventually be achieved, and the calculations can proceed in a stepwise fashion down the tube. 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

At present, black-box programs for solving initial value problems typically maintain fairly rigorous automatic control of the numerical errors. However, for multidimensional reacting-flow simulations (boundary value problems), error control is usually not automated, relying on the user performing the... 

In order to correctly model the different possible states of the system, it will be necessary to cover a large part of the accessible phase space, so either trajectories must be very long or we must use many initial conditions. There are many ways to solve initial value problems such as (2.1) combined with an initial condition z(0) = 5. The methods introduced here all rely on the idea of a discretization with a finite stepsize h, and an iterative procedure that computes, starting from zo =, a sequence zi,Z2,..., where z z(nh). The simplest scheme is certainly Euler s method which advances the solution from timestep to timestep by the formula ... 

This method is very efficient when it is applicable, and is one of the many integral transformations that are useful for solving initial value problems. Integral transforms are certain functions that are defined by an integral, such as... 

Generating two solutions to the homogeneous equation (y" = 0) with the requirement that the boundary conditions be independent. This is the only requirement we must meet to solve initial-value problems since the boundary conditions used in applying the... 

J. Ibinez, V. Hernandez, E. Arias and P. A. Ruiz, Solving Initial Value Problems For Ordinary Differential Equations by Two Approaches BDF And Piecewise-Linearized Methods, Computer Physics Communications, 2009, 180(5), 712-723. 

As explained in Sect. 2.1, a full description of the time-dependent progress of a chemical reaction system requires a mechanism containing not just reactants and products but also important intermediate species. The rate of consumption of the species within the mechanism can vary over many orders of magnitude depending on the species type. Radical intermediates, for example, usually react on quicker timescales than stable molecular species. This can lead to numerical issues when attempting to solve initial value problems such as that expressed in Eq. (5.1), since the variation in timescales can lead to a stiff differential equation system which may become numerically unstable unless a small time step is used or special numerical... 

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