MATLAB: Confusion on a new schema for solving ode

solve ode numerically new schema

Hello all, recently i've been working on an algorithm for solving ode, called QSS family, which was found by Prof.Ernesto Kofman. I want to run the algorithm on matlab code, but i'm not sure if the code is right. Can anyone help me to tell if it's right?
The code below is an example to solve the ode:, with and all equal 0 at .
clc
clear
tic
delta_q=1e-4;
q1=0;q2=0;
x1=0;x2=0;
t=0;delta_t=0;tfinal=20;
A=zeros(0,3);
n=0;
while (t<tfinal)
    Dx1=q2;
    Dx2=1-3*q1-4*q2;
    if (Dx1>0)
        delta_x1=delta_q+q1-x1;
    else
        delta_x1=delta_q-q1+x1;
    end 
   if (Dx2>0)
       delta_x2=delta_q+q2-x2;
   else
       delta_x2=delta_q-q2+x2;
   end
       delta_t1=delta_x1/abs(Dx1);
       delta_t2=delta_x2/abs(Dx2);
   if (delta_t1<delta_t2)
       delta_t=delta_t1;
       t=t+delta_t;
       x1=x1+Dx1*delta_t;
       x2=x2+Dx2*delta_t;
       q1=x1;
   else
       delta_t=delta_t2;
       t=t+delta_t;
       x1=x1+Dx1*delta_t;
       x2=x2+Dx2*delta_t;
       q2=x2;
   end
   n=n+1;
   A(n,1)=t;
   A(n,2)=x1;
   A(n,3)=x2;
%    A(n,4)=q1;
%    A(n,5)=q2;
end
A;
 figure(1)
 plot(A(:,1),A(:,2),A(:,1),A(:,3));
toc

Best Answer

Hello, 
function euler1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clc
clear
tic
delta_q=1e-4;
q1=0;q2=0;
x1=0;x2=0;
t=0;delta_t=0;tfinal=20;
A=zeros(0,3);
n=0;
while (t<tfinal)
    Dx1=q2;
    Dx2=1-3*q1-4*q2;
    if (Dx1>0)
        delta_x1=delta_q+q1-x1;
    else
        delta_x1=delta_q-q1+x1;
    end
    if (Dx2>0)
        delta_x2=delta_q+q2-x2;
    else
        delta_x2=delta_q-q2+x2;
    end
    delta_t1=delta_x1/abs(Dx1);
    delta_t2=delta_x2/abs(Dx2);
    if (delta_t1<delta_t2)
        delta_t=delta_t1;
        t=t+delta_t;
        x1=x1+Dx1*delta_t;
        x2=x2+Dx2*delta_t;
        q1=x1;
    else
        delta_t=delta_t2;
        t=t+delta_t;
        x1=x1+Dx1*delta_t;
        x2=x2+Dx2*delta_t;
        q2=x2;
    end
    n=n+1;
    A(n,1)=t;
    A(n,2)=x1;
    A(n,3)=x2;
    %    A(n,4)=q1;
    %    A(n,5)=q2;
end
figure(1)
close(1)
figure(1)
subplot(1,3,1)
hold on
axis([0,20 -0.05 0.35])
plot(A(:,1),A(:,2),'b')
plot(A(:,1),A(:,3),'r');
grid on
title('HS')
toc
whos
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
t=linspace(0,20,501);
x0=[0 0];
whos
[t,x]=ode45(@fun,t,x0);
whos
subplot(1,3,2)
hold on
axis([0,20 -0.05 0.35])
plot(t,x(:,1),'k')
plot(t,x(:,2),'k')
grid on
title('edo45')
subplot(1,3,3)
hold on
axis([0,20 -0.05 0.35])
plot(A(:,1),A(:,2),'b')
plot(A(:,1),A(:,3),'r');
plot(t,x(:,1),'k.')
plot(t,x(:,2),'k.')
grid on
title('HS, edo45')
    function [dxdt]=fun(~,x)
        x1p=x(2);
        x2p=1-3*x(1)-4*x(2);
        dxdt=[x1p; x2p];
    end
end
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