Adams-Bashforth 4th order

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LUCA SCARPELLI 2021-7-17

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https://ww2.mathworks.cn/matlabcentral/answers/880483-adams-bashforth-4th-order

编辑： LUCA SCARPELLI 2021-7-17

Hi everyone,

I'm trying to implement the AB4 to solve ODE's, and my code seems to be wrong as it's not compiling: I get the error message "Index exceeds the number of array elements (4)."

I tried to check the range of implementation for the code in the AB part (the second loop in the code). I used R-K4 to find the initial condition, as they are four.

Here is my code:

function adams_bashforth_method(a,b,s,f,y0)
% s is the number of subinterval for the interpolation
k=3;                    % steps-1
x=linspace(a,b, s+1);   %network set creation
h=1/(s+1);              %discretization of the network set
u=zeros(1,s+1);         %network function creation
u(1)=y0;                %initial condition of Cauchy problem
for i=1:k
    k_1=f(x(i),u(i));                            %k_1 
    k_2=f(x(i)+h./2,u(i)+((h./2).*k_1));         %k_2
    k_3=f(x(i)+h./2,u(i)+((h./2).*k_2));         %k_3
    k_4=f(x(i)+h,u(i)+(h.*k_3));                 %k_4
    u(i+1)=u(i)+(h/6).*(k_1+2.*(k_2+k_3)+k_4);   %Runge-Kutta
end
%Now that I know the initial four condition, as I set the step constant, I
%can evaluate the others u's by AB4 formula
for i=4:s 
    u(i+1)=u(i)+(h./24).*(55.*(f(x(i+3),u(i+3)))-59.*(f(x(i+2),u(i+2)))+37.*(f(x(i+1),u(i+1)))-9.*(f(x(i),u(i))));
end
plot(x,u,'r');
title('Adams Bashforth 4th order method')
hold on
end