A problem that the Ode45 solver did not finish the computations

Asked by Xuan Ling Zhang on 21 Oct 2019

Latest activity Commented on by Xuan Ling Zhang on 22 Oct 2019

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Hi, i have a problem when use the Ode45 slover. Hope someone can help me, thank you in advance！！！ The problem will be described step by step as follows：

I have a boundary-value problem expressed by

I use the bvp5c solver to deal with this problem. The code is given as：

function main

solinit = bvpinit([0:0.001:L],zeros(1,6));

sol = bvp5c(@Fun,@bvpf,solinit);

plot(sol.x,sol.y(3,:))

end

function dy = Fun(x,y)

f0=230;

L=0.5;

G0=6.5282; Gw1 =1.8855e+006; Gw2 =-1.3920e-012; Gw3 =6.5986e-008; Gw4 =4.4582e-012;Gw5 =6.3222e+007;

Gz1 =1.4801e+004; Gz2 =-0.3122; Gz3 =4.5925e-014; f0=230;

dy = zeros(6,1);

dy(1) = y(2);

dy(2) = Gz1*(y(1)+y(4))-Gz2*y(6)-Gz3*y(4)*y(5);

dy(3) = y(4);

dy(4) = y(5);

dy(5) = y(6);

dy(6) = Gw1*(y(2)+y(5))+Gw2*y(5)^2+Gw3*y(4)^2+Gw3*y(4)*y(1)+Gw4*y(2)*y(5)+ ...

Gw5*y(4)*y(4)*y(5)+f0/G0*sin(pi*x/L);

end

function dy = bvpf(ya,yb)

dy = [

ya(1)

yb(1)

ya(3)

yb(3)

ya(4)

yb(4)

];

end

According to the code above, the solution is obtained readily and validated by other numerical method. The solution at x and L is obtained as

sol.y(:,1)=[0; -0.0022477; 0; 0; 0.0069115; -5.607309];

sol.y(:,end)=[0; -0.0022477; 0; 0; 0.0069115; -5.607309];

Now, i use the Ode45 solver to re-compute the problem, where the initial value is set to be sol.y(:,1)：

function main

options = odeset('RelTol',1e-9,'AbsTol',1e-12);

span=[0 L]

[x,y1]=ode45(@Fun,span,sol.y(:,1),options);

end

Obviously, the solution y1(:,end) should be equal to sol.y(:,end), but ode45 solver did not even finish the computation. I don't know why this happened.

3 Comments

David Wilson on 21 Oct 2019

Did you forget the three dots (line continuation) in the ODE function ?

 dy(6) = Gw1*(y(2)+y(5))+Gw2*y(5)^2+Gw3*y(4)^2+Gw3*y(4)*y(1)+Gw4*y(2)*y(5) + ...
         Gw5*y(4)*y(4)*y(5)+f0/G0*sin(pi*x/L);

In any case, the solving the ODE could be (numerically) unstable, whereas solving the BVP is actually OK.

Xuan Ling Zhang on 21 Oct 2019

Dear Wilson,

sorry, I just forget the three dots in the above edit, and i'm sure that there is no problem on the ODE function in my Matlab m file.

Xuan Ling Zhang on 22 Oct 2019

Dear Wilson,

Thanks for your good suggestion in the comments below, according to which i will try it.

Best wishes.

Yours sincerely,

1 Answer

Link

Answer by John D'Errico on 21 Oct 2019

Edited by John D'Errico on 21 Oct 2019

The answer is (probably) simple. I need point out only a couple of lines of your code that tells me it is so. (In fact, I was almost certain of it as soon as I saw your comment that ODE did not finish the computation, without even needing to look at any code at all.)

https://en.wikipedia.org/wiki/Stiff_equation

You very likely have a stiff ODE.

Usually, this seems to happen when you have different processes acting in the system you are modeling with widely varying time scales. Many chemical and biological systems seem to fall into this class of problem.

Consider these lines of code from your problem:

Gz1 =1.4801e+004; Gz2 =-0.3122; Gz3 =4.5925e-014; f0=230;
...
dy(2) = Gz1*(y(1)+y(4))-Gz2*y(6)-Gz3*y(4)*y(5);
...
G0=6.5282; Gw1 =1.8855e+006; Gw2 =-1.3920e-012; Gw3 =6.5986e-008; Gw4 =4.4582e-012;Gw5 =6.3222e+007;
...
dy(6) = Gw1*(y(2)+y(5))+Gw2*y(5)^2+Gw3*y(4)^2+Gw3*y(4)*y(1)+Gw4*y(2)*y(5)
             +Gw5*y(4)*y(4)*y(5)+f0/G0*sin(pi*x/L);

So the various coefficients are either tiny, or quite large in terms of double precision. The result is that ODE45 will find it necessary to use an incredibly tiny time step to resolve those processes, but only when they become significant.

Essentially, ODE45 just gives up when the necessary time step becomes so small that no effort can be made in double precision arithmetic. (It should have told you that.)

The answer is USUALLY to switch to a stiff solver. In MATLAB, that means to use one of the solvers ode15s or ode23s. (Any solver with a name in it that ends in s.)

6 Comments

Show 3 older comments

Steven Lord on 21 Oct 2019

When I try setting one additional option to see what the solution looks like:

options = odeset('RelTol',1e-9,'AbsTol',1e-12, 'OutputFcn', @odeplot);

I see your function falls off a cliff at around 0.02655434. By the time I stopped it at x = 0.0265543425513827, the value of the third component of the solution was around -0.00688 and the value of the sixth was -3.97e34.

Xuan Ling Zhang on 22 Oct 2019

Dear Lord,

Yes, this is what happened that i don't know why. Do you have some suggestions？

Xuan Ling Zhang on 22 Oct 2019

Dear Wilson, the first scheme of 'integrate the system backwards' that i have tried but fails.

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