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assign2 - Handout Six Part II November 3 2011 1...

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Handout Six Part II November 3, 2011 1 Supplementary Notes : Solving an Initial Value Problem for an Ordinary Differential Equation It is not the aim of these notes to teach a course on the numerical solution of ordinary differential equations (ODE’s). However you do need to be able to understand the methodology behind the subject so you are able to ‘code up’ a simple method. A numerical method for solving an ordinary differential equation is a procedure that produces approximate solutions to the differential equation at progressive points along the independent variable. It does this using the given differential equation and an accompanying initial condition as a starting point. You will only need to write Fortran programs for problems of the form; y = dy dx = f ( x, y ) differential equation (1) y 0 = y ( x 0 ) initial condition (2) The first equation is simply the differential equation, it gives us the ‘rate of change’ of the dependent variable y with respect to the independent variable x . The second equation tells us that at a known point of the independent variable x the value of the function y ( x ) is known and is y 0 . With these initial conditions we can attempt to produce a numerical solution to the ODE over a range of values in x . This is done by starting at x 0 where we know the solution y 0 as it is given, and making an approximation for y at x = x 0 + h . The differential equation is then used, along with a small step h in the x direction to give an estimate δy of the change in y that the small step h in x would produce. There are many different methods for solving ODE’s, some very much more complicated than others. In this handout we will look at seven of the simpler methods around. The Euler method, Heun’s method, Nystrom’s and a third order Runge-Kutta method. 1.1 The Order of a Numerical Method In general the higher the order of a numerical method the more accurate it is. The order of a numerical method is given as an positive integer. Euler’s method is of order one, so it the lowest possible order you can have. If the exact solution of an ordinary differential equation is a polynomial of order k then the numerical solution and the exact solution will be identical (except for computational truncation error) if the numerical method is of order k or above. 1.2 Simple Numerical Methods First we will look at a few of the simplest of numerical methods for ODE’s. Then a couple of methods well known group of ‘Runge-Kutta’ based solutions. Finally a couple of ‘predictor-corrector’ methods. 1.2.1 Euler’s Method Euler’s method is probably the most simple of all numerical methods for solving ordinary differential equations numerically. Given the point ( x n , y n ) and the gradient f n at that point, where f n = f ( x n , y n ), 1
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Euler’s method makes the assumption that over the sufficiently small step, h , in x the function is ap- proximately linear. This allows us to write down an approximation for y n +1 at x n +1 = x + h .
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