Ordinary Differential Equations. Monday, October 10, 11
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1 Ordinary Differential Equations Monday, October 10, 11
2 Problems involving ODEs can always be reduced to a set of first order differential equations. For example, By introducing a new variable z, this can be rewritten as: This exemplifies the procedure for an arbitrary ODE. The usual choice for the new variables is to let them just be derivatives of each other and of the original variable. Monday, October 10, 11
3 Let common sense be your guide: If you find that the original variables are smooth in a solution, while your auxiliary variables are doing crazy things, then figure out why and choose different auxiliary variables. Monday, October 10, 11
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5 Lorenz Oscillator Monday, October 10, 11
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8 Non-linearities may exist Monday, October 10, 11
9 Is a problem involving ODEs completely specified by its equations?
10 Is a problem involving ODEs completely specified by its equations? NO! Even more crucial in determining how to attack the problem numerically is the nature of the boundary conditions. Boundary conditions are algebraic conditions on the values of the functions yi in: Monday, October 10, 11
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12 In general, the boundary conditions can be specified at discrete points, but do not hold between those points, i.e. are not preserved automatically by the differential equations. Boundary conditions may be as simple as requiring that certain variables have certain numerical values, or as complicated as a set of non-linear algebraic equations among the variables. Monday, October 10, 11
13 Usually it is the nature of the boundary conditions that determines which numerical methods will be feasible. Boundary conditions divide into two broad categories. Monday, October 10, 11
14 The underlying idea of any routine for solving the initial value problem is always this: Rewrite the dy s and dx s in as finite steps in y and x, and multiply the equations by x. This gives algebraic formulas for the change in the functions when the independent variable, x, is stepped by one stepsize x. In the limit of making the stepsize very small ( x 0), a good approximation to the underlying differential equation is achieved. Monday, October 10, 11
15 The underlying idea of any routine for solving the initial value problem is always this: Rewrite the dy s and dx s in as finite steps in y and x, and multiply the equations by x. This gives algebraic formulas for the change in the functions when the independent variable, x, is stepped by one stepsize x. In the limit of making the stepsize very small ( x 0), a good approximation to the underlying differential equation is achieved. Monday, October 10, 11
16 Literal implementation of this procedure results in Euler s method, which is however not recommended for any practical use. Euler s method is conceptually important, however; one way or another practical methods all come down to the same idea: Add small increments to your functions corresponding to derivatives (right-hand sides of the equations) multiplied by stepsizes. Monday, October 10, 11
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20 Homework 4 Starting from the conditions below write a Euler integration for the simple harmonic oscillator to produce the plot on the right. Why are the solutions diverging? Location Harmonic Oscillator Time Monday, October 10, 11
21 Homework 4 Solve the same ODEs by calling ODE23, ODE45, ODE113 overlay on the Euler solutions like on the plot to the Location Harmonic Oscillator right Time Monday, October 10, 11
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23 As the step size increased so did the error! Why? 4 Harmonic Oscillator 3 2 Location Time Monday, October 10, 11
24 As the step size increased so did the error! Why? 4 Harmonic Oscillator 3 2 Location Time The slope changes over our time step, but in Euler s Monday, October 10, 11 method we assume it is constant.
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28 ODE Solvers There are three types of solvers: Runge-Kutta Methods Richardson extrapolation and in particular the Burlish-Stoer method Predictor corrector methods Monday, October 10, 11
29 ODE Solvers
30 Which method should I use?
31 Which method should I use?
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39 Exercise Look at Matlab help: doc ode23 Do the three examples Open the document odes.pdf and read about the Lorenz attractor in section 7.8 Look at the Lorenz attractor example lorenzgui.m Read about stiff systems in section 7.9 Read chapter 17 of Numerical Recipes Monday, October 10, 11
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