Day5 - CE 561 Lecture Notes Fall 2009 Day 5: Numerical...

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CE 561 Lecture Notes Fall 2009 p. 1 of 10 Day 5: Numerical Methods for Integrating Rate Equations Numerical methods for integrating rate equations Most situations that we encounter involve reaction mechanisms that are not a network of first order reactions, but involve some higher order reactions (or non-isothermal conditions, etc.) and therefore lead to nonlinear rate equations. In general, we will not be able to solve these systems of nonlinear ODEs analytically. We may be able to make approximations (like the pseudo- steady-state or partial equilibrium approximations) that allow us to solve the rate equations, or we may be able to linearize the equations by assuming that some species concentrations are approximately constant. Often, though, we will have to resort to numerical solutions of the ODE’s. The differential equations arising from descriptions of chemical kinetics sometimes present some special difficulties. This set of notes is intended to introduce you to numerical methods for initial value problems (first order ODE's), with some of these difficulties in mind. The goal is not for you to become an expert in numerical methods, but simply to provide enough understanding of these methods and the issues involved in solving nonlinear initial value problems so that you can intelligently select and use software packages written by others. The explicit Euler method: This method is the simplest technique for numerically solving an initial value problem and can be regarded as the prototypical method. For the first-order ordinary differential equation dy dt f y = ( ) We approximate the derivative as dy dt y t y t t y t t = + ( ) ( ) where t is some small time step. So, we have the approximate equation ( ) () yt t fy t +∆ − = Then we solve this for y ( t + t ) to get y t t y t f y t t ( ) ( ) ( ( )) + = + For example, if we have the initial value problem dy dt y y t = = = 2 0 1 , with initial condition ( ) and we take time steps of 0.1, then the integration procedure is y y y ( . ) . . ( . ) . ( . ) . . ( . ) . ( . ) . . 01 1 1 11 0 2 1221 0 3 1370 2 2 2 = + × = = + × = = + × = and so on.
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CE 561 Lecture Notes Fall 2009 p. 2 of 10 The advantage of this method is that it is intuitive and simple to implement. The disadvantages are that it is relatively inaccurate (compared to more sophisticated methods), and it may require very small time steps. It is called an explicit method, because the time derivative at a given time step is explicitly evaluated in terms of the value of the function at previous time steps (and not in terms of the function value at the present time step). That is, given y ( t ) and f ( y ), we can explicitly evaluate y ( t + t ). Stability of the numerical method It can be shown that the explicit Euler method, like many other explicit methods, only works for sufficiently small time steps. For a time step larger than some critical value, the method becomes unstable and diverges from the true solution.
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Day5 - CE 561 Lecture Notes Fall 2009 Day 5: Numerical...

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