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Math Help - Fourth Order Runge-Kutta for two-simultaneous 1st ODE

  1. #1
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    Fourth Order Runge-Kutta for two-simultaneous 1st ODE

    I am having real trouble in creating a fourth order Runge-Kutta method for a system of two simultaneous first order ordinary differential equations.

    It is supposed to model the wake off an aircraft wing, in terms of it's x and y coordinates.

    Fourth Order Runge-Kutta for two-simultaneous 1st ODE-wake.png
    The wing has been split into 10 parts, i.e. 5 on each side of the wing. And the program should give 10 values of x and y for each time step (i.e. 10 lots of 500 time steps in this case)
    Any help with this problem would be gratefully appreciated..My attempted matlab program is below, but i fear ive got lost....


    clc
    m=10;
    m2=m/2;
    dx=1/m;
    NMAX=500;
    nfreq=1;
    t=0;
    x(1)=-1+dx;
    ndt=0;
    h=1;
    dt=1/25/m;

    for
    J=2:m;
    x(J)=x(J-1)+2*dx;


    end
    for
    J=1:m2;
    y(J)=0;
    g(J)=((1-((x(J)+dx)^2))^0.5)-((1-((x(J)-dx)^2))^0.5);

    end

    m2p1=m2+1;

    for
    J=m2p1:m;
    y(J)=0;
    g(J)=-g(m+1-J);

    end


    U=0;
    V=0;
    XI=-0.9;
    YI=0;

    NDT=0;
    T=0;

    while
    NMAX>NDT
    NDT=NDT+1;
    T=T+dt;

    for
    J=1:10;
    if J~=I;

    U1=U+g(J)*(YI-Y(J))/((XI-X(J))^2 + (YI-Y(J))^2);
    V1=V-g(J)*(XI-X(J))/((XI-X(J))^2 + (YI-Y(J))^2);
    D1X= dt*U1;
    D1Y= dt*V1;
    X2=XI+D1X/2;
    Y2=YI+D1Y/2;
    U2=U+g(J)*(Y2-Y(J))/((X2-X(J))^2 + (Y2-Y(J))^2);
    V2=V-g(J)*(X2-X(J))/((X2-X(J))^2 + (Y2-Y(J))^2);
    D2X= dt*U2;
    D2Y= dt*V2;
    X3=XI+D2X/2;
    Y3=YI+D2Y/2;
    U3=U+g(J)*(Y3-Y2)/((X3-X2)^2 + (Y3-Y2)^2);
    V3=V-g(J)*(X3-X2)/((X3-X2)^2 + (Y3-Y2)^2);
    D3X= dt*U3;
    D3Y= dt*V3;
    X4=XI+D3X;
    Y4=YI+D3Y;
    U4=U+g(J)*(Y4-Y3)/((X4-X3)^2 + (Y4-Y3)^2);
    V4=V-g(J)*(X4-X3)/((X4-X3)^2 + (Y4-Y3)^2);
    D4X= dt*U4;
    D4Y= dt*V4;

    X(J+1) = X(J)+(D1X+2*D2X+2*D3X+D4X)/6.0;
    Y(J+1) = Y(J)+(D1Y+2*D2Y+2*D3Y+D4Y)/6.0;

    X(I)=X(J+1);
    Y(I)=Y(J+1);
    end
    end
    for
    I=m2p1:m;
    X(I)=-X(m+1-I);
    Y(I)=-Y(m+1-I);

    end

    X(1:10)
    Y(1:10)

    end

    g(1:10)


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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by p123nky View Post
    I am having real trouble in creating a fourth order Runge-Kutta method for a system of two simultaneous first order ordinary differential equations.

    It is supposed to model the wake off an aircraft wing, in terms of it's x and y coordinates.

    Click image for larger version. 

Name:	wake.png 
Views:	120 
Size:	13.9 KB 
ID:	19559
    The wing has been split into 10 parts, i.e. 5 on each side of the wing. And the program should give 10 values of x and y for each time step (i.e. 10 lots of 500 time steps in this case)
    Any help with this problem would be gratefully appreciated..My attempted matlab program is below, but i fear ive got lost....



    Introduce the state vector {\bf{X}}=(X_1, X_2,...,X_m,Y_1, Y_2,...,Y_m)^t.

    Now write a function deriv(t, {\bf{X}}) that returns {\bf{X}}'.

    You should now be able to use the update equations for the scalar RK4 on these vectors without modification.

    CB
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