Page 284 - Applied Numerical Methods Using MATLAB

PREDICTOR–CORRECTOR METHOD 273

function [t,y] = ode_ABM(f,tspan,y0,N,KC,varargin)
%Adams-Bashforth-Moulton method to solve vector d.e. y’(t) = f(t,y(t))
% for tspan = [t0,tf] and with the initial value y0 and N time steps
% using the modifier based on the error estimate depending on KC = 1/0
if nargin < 5, KC = 1; end %with modifier by default
if nargin<4|N<=0, N= 100; end %default maximum number of iterations
y0 = y0(:)’; %make it a row vector
h = (tspan(2) - tspan(1))/N; %step size
tspan0 = tspan(1)+[0 3]*h;
[t,y] = rk4(f,tspan0,y0,3,varargin{:}); %initialize by Runge-Kutta
t = [t(1:3)’ t(4):h:tspan(2)]’;
for k = 1:4, F(k,:) = feval(f,t(k),y(k,:),varargin{:}); end
p = y(4,:); c = y(4,:); KC22 = KC*251/270; KC12 = KC*19/270;
h24 = h/24; h241 = h24*[1 -5 19 9]; h249 = h24*[-9 37 -59 55];
for k = 4:N
p1 = y(k,:) +h249*F; %Eq.(6.4.8a)
m1 = pk1 + KC22*(c-p); %Eq.(6.4.8b)
c1 = y(k,:)+ ...
h241*[F(2:4,:); feval(f,t(k + 1),m1,varargin{:})]; %Eq.(6.4.8c)
y(k + 1,:) = c1 - KC12*(c1 - p1); %Eq.(6.4.8d)
p = p1; c = c1; %update the predicted/corrected values
F = [F(2:4,:); feval(f,t(k + 1),y(k + 1,:),varargin{:})];
End

6.4.2 Hamming Method

function [t,y] = ode_Ham(f,tspan,y0,N,KC,varargin)
% Hamming method to solve vector d.e. y’(t) = f(t,y(t))
% for tspan = [t0,tf] and with the initial value y0 and N time steps
% using the modifier based on the error estimate depending on KC = 1/0
if nargin < 5, KC = 1; end %with modifier by default
if nargin<4|N<=0,N = 100; end %default maximum number of iterations
if nargin < 3, y0 = 0; end %default initial value
y0 = y0(:)’; end %make it a row vector
h = (tspan(2)-tspan(1))/N; %step size
tspan0 = tspan(1)+[0 3]*h;
[t,y] = ode_RK4(f,tspan0,y0,3,varargin{:}); %Initialize by Runge-Kutta
t = [t(1:3)’ t(4):h:tspan(2)]’;
for k = 2:4, F(k - 1,:) = feval(f,t(k),y(k,:),varargin{:}); end
p = y(4,:); c = y(4,:); h34 = h/3*4; KC11 = KC*112/121; KC91 = KC*9/121;
h312 = 3*h*[-1 2 1];
for k = 4:N
p1 = y(k - 3,:) + h34*(2*(F(1,:) + F(3,:)) - F(2,:)); %Eq.(6.4.9a)
m1 = p1 + KC11*(c - p); %Eq.(6.4.9b)
c1 = (-y(k - 2,:) + 9*y(k,:) +...
h312*[F(2:3,:); feval(f,t(k + 1),m1,varargin{:})])/8; %Eq.(6.4.9c)
y(k+1,:) = c1 - KC91*(c1 - p1); %Eq.(6.4.9d)
p = p1; c = c1; %update the predicted/corrected values
F = [F(2:3,:); feval(f,t(k + 1),y(k + 1,:),varargin{:})];
end