Explanation The Matlab program run with the function replaced by the unstable two-step method. The resulting approximations follow: %Multistep method program % Inputs: time interval inters, ic = [y0] initial condition, number of steps n, % s=number of (multi)steps, e.g. 2 for 2-step method % Output: time steps t, solution y % Calls a multistep method such as ab2step.m % function [t,y]=exmultistep(inter,ic,n,s) h=(inter(2)-inter(1))/n; % Start-up phase y(1,:)=ic;t(1)=inter(1); for i=1:s-1 % start-up phase, using one-step method t(i+1)=t(i)+h; y(i+1,:)=trapstep(t(i),y(i,:),h); f(i,:)=ydot(t(i),y(i,:)); end for i=s:n % multistep method loop t(i+1)=t(i)+h; f(i,:)=ydot(t(i),y(i,:)); y(i+1,:)=ab2step(t(i),i,y,f,h); end plot(t,y) function y=trapstep(t,x,h) %one step of the Trapezoid Method z1=ydot(t,x); g=x+h*z1; z2=ydot(t+h,g); y=x+h*(z1+z2)/2; function z=ab2step(t,i,y,f,h) %one step of the Adams-Bashforth 2-step method z=y(i,:)+h*(3*f(i,:)/2-f(i-1,:)/2); function z=unstable2step(t,i,y,f,h) %one step of an unstable 2-step method z=-y(i,:)+2*y(i-1,:)+h*(5*f(i,:)/2+f(i-1,:)/2); function z=weaklystable2step(t,i,y,f,h) %one step of a weakly-stable 2-step method z=y(i-1,:)+h*2*f(i,:); function z = ydot(t,y) % IVP z = t*y+t^3; % initialize f = inline('y/x-y^2/x^2','x','y'); xrange=[1,2]; h=0.1; x=1:h:2; n = (xrange(2)-xrange(1))/h; y(1) = 1; % generate starting estimates using Runge-Kutta for i = 1 k1 = f(x(i), y(i)); k2 = f(x(i) + h/2, y(i) + h/2*k1); k3 = f(x(i) + h/2, y(i) + h/2*k2); k4 = f(x(i) + h, y(i) + h*k3); y(i+1) = y(i) + h/6*(k1 + 2*k2 + 2*k3 + k4); x(i+1) = x(i) + h; end % iterate for i = 3:n+1 % Adams-Moulton -- *correct* y(i) = y(i-1) + h/12*(5*f(x(i),y(i)) + 8*f(x(i-1),y(i-1))- f(x(i-2),y(i-2))); end [t,y]=exmultistep([0 1],1,20,2); (a) t i w i error 0.0 0.0000 0.0000 0.1 0.0050 0.0002 0.2 0.0213 0.0002 0.3 0.0493 0.0005 0.4 0.0916 0.0002 0.5 0.1474 0.0013 0.6 0.2222 0.0001 0.7 0.3105 0.0032 0.8 0.4276 0.0020 0.9 0.5510 0.0086 1.0 0.7283 0.0100 (b) t i w i error 0

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Author: Sauer, Tim

Publisher: Pearson,

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