Adams 4阶4步显式方法 MATLAB
function [wi, ti] = adams_pc4 ( RHS, t0, x0, tf, N )
%ADAMS_PC4 approximate the solution of the initial value problem %
% x'(t) = RHS( t, x ), x(t0) = x0 %
% using the Adams fourth-order predictor / corrector scheme % - this routine will work for a system of first-order % equations as well as for a single equation
%
% the classical fourth-order Runge-Kutta method is used to % initialize the predictor / corrector scheme %
%
% calling sequences:
% [wi, ti] = adams_pc4 ( RHS, t0, x0, tf, N ) % adams_pc4 ( RHS, t0, x0, tf, N )
%
% inputs:
% RHS string containing name of m-file defining the % right-hand side of the differential equation; the
% m-file must take two inputs - first, the value of % the independent variable; second, the value of the % dependent variable
% t0 initial value of the independent variable % x0 initial value of the dependent variable(s) % if solving a system of equations, this should be a
% row vector containing all initial values % tf final value of the independent variable % N number of uniformly sized time steps to be taken to
% advance the solution from t = t0 to t = tf %
% output:
% wi vector / matrix containing values of the approximate
% solution to the differential equation % ti vector containing the values of the independent % variable at which an approximate solution has been % obtained
%
neqn = length ( x0 );
ti = linspace ( t0, tf, N+1 );
wi = [ zeros( neqn, N+1 ) ];
wi(1:neqn, 1) = x0';
h = ( tf - t0 ) / N;
oldf = zeros(3,neqn);
%
% generate starting values using classical 4th order RK method
% remember to save function values
%
for i = 1:3
oldf(i,1:neqn) = feval ( RHS, t0, x0 );
k1 = h * oldf(i,:);
k2 = h * feval ( RHS, t0 + h/2, x0 + k1/2 );
k3 = h * feval ( RHS, t0 + h/2, x0 + k2/2 );
k4 = h * feval ( RHS, t0 + h, x0 + k3 );
x0 = x0 + ( k1 + 2*k2 + 2*k3 + k4 ) / 6;
t0 = t0 + h;
wi(1:neqn,i+1) = x0';
end;
%
% continue time stepping with 4th order Adams Predictor / Corrector
%
for i = 4:N
fnew = feval ( RHS, t0, x0 );
xtilde = x0 + (h/24) * ( 55*fnew - 59*oldf(3,:) + 37*oldf(2,:) -
9*oldf(1,:) );
fnew1 = feval ( RHS, t0+h, xtilde );
x0 = x0 + (h/24) * ( 9*fnew1 + 19*fnew - 5*oldf(3,:) + oldf(2,:) );
oldf(1,1:neqn) = oldf(2,1:neqn);
oldf(2,1:neqn) = oldf(3,1:neqn);
oldf(3,1:neqn) = fnew;
t0 = t0 + h;
wi(1:neqn,i+1) = x0';
end;
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