I'm using NDSolve to solve a set of coupled differential equation which depend on a variable x. I noticed that when I set the range of x from a small value to a large value, I obtain solutions. But when I do the opposite I get a problem!
For example when I consider this,
t1 = 3000; t2 = 4*^16; f = 1/(16 Pi^2); NDSolve[{y'[x] == f/x * 16 y[x]^3, y[t1] == 0.37}, y, {x, t1, t2}, Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4}]; No errors and I can find:
y[t2] /. % {0.920385} Then I use this value of y[t2] as the initial condition for the following case, where I swap the range of x:
t1 = 3000; t2 = 4*^16; f = 1/(16 Pi^2); NDSolve[{y'[x] == f/x * 16 y[x]^3, y[t2] == 0.92}, y, {x, t2, t1}, Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4}] I get the error:
NDSolve::ndsz: At x == 4.`*^16, step size is effectively zero; singularity or stiff system suspected. I tried to switch off the StiffnessTest to see if the problem goes away, which would mean that it's a stiffness problem, but the error message still shows. Also tried to use:
Method -> {StiffnessSwitching, Method -> {ExplicitRungeKutta, Automatic}} And it didn't work either.
Considering that this is a typical differential equation encountered in physics (renormalization group equations) I'm pretty sure that there should be no singularity at x = t2, and I should be able to solve from the high-scale to the low-scale.
Any insights on why this is happening and how to deal with it?
