Asked by Xuan Ling Zhang
on 21 Oct 2019 at 7:34

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.

Answer by John D'Errico
on 21 Oct 2019 at 8:15

Edited by John D'Errico
on 21 Oct 2019 at 8:16

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.)

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.)

Steven Lord
on 21 Oct 2019 at 22:56

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 at 1:41

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 at 2:19

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

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## David Wilson (view profile)

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## Xuan Ling Zhang (view profile)

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## Xuan Ling Zhang (view profile)

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