Computational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations

Transcription

1 /0/0 Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami Dept. of Computer Science Several slides adapted from Profs. Dianne O Lear and Eric Sandt TAMU Higher Order methods: Complications All we have availale to us is a lackox routine that returns d/dx=f(x) Don t have higher order derivatives Also chain rule complicates things d /dx =df(x)/dx=( f/ ) d/dx+df/dx =f f+f x So we need some wa to estimate f x and f at initial point (x 0 0 ) from f Use multivariate mean value theorem Approximate derivatives estimating function values near

2 /0/0 The initial conditions are: d dx f x x 0 0 To derive method we use the Talor series expansion including nd order terms h d x d x x x h dx! dx n n n n n n Expand the derivatives: d dx d dx d dx f x f f f f f The Talor series expansion ecomes n n hf h fx f f Have expressed second derivative in terms of st derivatives of f x x

3 /0/0 Look for a formula of the tpe ak k k hf x n n n n k hf x h k n n Goal is to find some intermediate points that allow us to approximate the second derivative Specification of a and provides the formula Look at formula we want for Runge-Kutta n n ahf hf xn h n hf Perform a multivariate Talor series expansion of the function Expand and group terms f x h hf f hf hf f n n x ahf h f hf hf f n n x n a hf h f x h f f

4 /0/0 4 Compare with the Talor series n n a hf h f h f f a 4 Unknowns Equations n n x hf h f f f The Talor series coefficients ( equations/4 unknowns) If ou select a as If ou select a as Note: These coefficient would result in a modified Euler or Midpoint Method a a a

5 /0/0 Runge-Kutta Method ( nd Order) Example Consider d dx The initial condition is: The step size is: h 0. Exact Solution 0 x Use the coefficients a Runge-Kutta Method Example ( nd Order) The values are k hf x i i k hf x h k i i k k i i 5

6 /0/0 Runge-Kutta Method ( nd Order) Example The values are similar to that of the Modified Euler also a second order method k Estimate Solution k Exact Error x n n ' n h' n * n+ *' n+ h(*' n+ ) Runge-Kutta Method Order) Example [] ( nd The values are a k hf x i i k hf x h k i i ak k i i 6

7 /0/0 Runge-Kutta Method Order) Example [] ( nd The values are k Estimate Solution k Exact Error x n n ' n h' n * n+ *' n+ h(*' n+ ) Exact Fourth order Runge-Kutta method n+ = n +/6(k +k +k +k 4 ) k =hf(x) k =h(f(x+h/ +/ k ) k =h(f(x+h/ +/ k ) k 4 =h(f(x+h+k )) 7

8 /0/0 4 th -order Runge-Kutta Method f f 4 f f f f 6 f f f f 4 x i x i + h/ x i + h Volterra example Write a function in standard form function f = rafox(t) % Computes ' for the Volterra model. % () is the numer of raits at time t. % () is the numer of foxes at time t. gloal alpha % interaction constant t % a print statement just so we can see how fast % the progress is and what stepsize is eing used f() = *() - alpha*()*(); f() = -() + alpha*()*(); 8

9 /0/0 Stud its solution for various values of encounter % Run the rait-fox model for various values of % the encounter parameter alpha plotting each % solution. gloal alpha for i=:-:0 alpha = 0^(-i) [t] = ode45('rafox'[0:.:] [00]); plot(t(:)'r't(:)''); legend('raits''foxes') title(sprintf('alpha = %f'alpha)); pause end Runge-Kutta Method Example (4 th Order) Consider d dx The initial condition is: The step size is: x h 0. Exact Solution x x x e 0 9

10 /0/0 The 4 th Order Runge-Kutta The example of a single step: k k k k 4 h n f x 0. f h f x h f x h f x h k 0. f n 6 h h 0. f k k k k k 0. f k k / Runge-Kutta Method Example (4 th Order) The values for the 4 th order Runge-Kutta method x f(x) k f k f k f 4 k 4 Change Exact

11 /0/0 The 4 th Order Runge-Kutta The step sizes are: k k k h The next step would e: k k k k x h x h x x h x 4 Stailit It turns out that explicit methods are not ver stale This means that the solution ma oscillate if we use large time steps So if we wish to integrate over a large interval and we need to take man small steps to achieve accurac man function evaluations are needed. Implicit methods are usuall more stale

12 /0/0 Implicit Methods There are second set of multi-step methods which are known as implicit methods. implicit => not directl revealed Here it means that the value of the function at the later time is not provided in an explicit formula ut in an equation Since future data is used an iterative method must e used to iterate an initial guess to convergence Could use Runge-Kutta or Adams Bashforth to start the initial value prolem.