Consider the initial value problem { y ′ = 1 − y 2 , t &gt; 0 y ( 0 ) = e − 1 e + 1 . Is there a way I find a limit on the mesh-size h such that the explicit Euler method is stable? Do I need to solve the IVP to find it?Thanks in advance.Edit: After looking through a number of textbooks and online resources I can see that they apply a method to a linear test problem and then solve | 1 + h λ | ⩽ 1. Is there any way I can do something similar to solve for h in this nonlinear case.

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Consider the initial value problem      {                                        y            ′                    =          1          −                      y            2                    ,                    t          &amp;gt;          0                                      y          (          0          )          =                                    e              −              1                                      e              +              1                                .                        Is there a way I find a limit on the mesh-size   h such that the explicit Euler method is stable? Do I need to solve the IVP to find it?Thanks in advance.Edit: After looking through a number of textbooks and online resources I can see that they apply a method to a linear test problem and then solve       |    1  +  h  λ      |    ⩽  1. Is there any way I can do something similar to solve for   h in this nonlinear case.

elulamami · Accepted Answer

If you have an explicit global Lipschitz constant, then you can go through the linear development in more or less the same manner, but when you don&#039;t that won&#039;t work. And here there is no global Lipschitz constant at all, much less one you can compute.That said here you can view things in a rather elementary manner. The point is that the true IVP stays in   (  −  1  ,  1  ) if it begins there, so you&#039;ll have stability if your method does the same. Since       y    ′    &amp;gt;  0 while you are in   (  −  1  ,  1  ), that means that as you go forward in time you want   y  +  h  (  1  −      y    2    )  &amp;lt;  1 when   y is any number between −1 and 1. From here you can readily infer a condition on   h. If you want backward-in-time stability (which the true IVP also has) then you want   y  −  h  (  1  −      y    2    )  &amp;gt;  −  1, which winds up resulting in the same condition on h in the end.This is not much different from the stability analysis of       y    ′    =  −  y with explicit Euler, where everything is OK going forward in time as long as you avoid crossing the   x-axis. Note that like that simpler problem, what I&#039;ve said here does not imply you will have blowup if the method exits   (  −  1  ,  1  ). In particular, you could have oscillation around   y  =  1, or you could just decay to   y  =  1 from above instead of below. In other words the condition we generate here is sufficient, not necessary, for stability.

Consider the initial value problem {y′=1−y^2,t>0, y(0)=e−1/e+1}. Is there a way I find a limit on the mesh-size h such that the explicit Euler method is stable? Do I need to solve the IVP to find it?

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