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I came across the method of multiple-scale analysis and was intrigued, because I am trying to solve a linear ODE with multiple characteristic timescales. When I apply the method as described, say, here, I find that the zeroth-order solution is the exact solution! This is exactly what I don't want, because conventional methods have not been successful in integrating this equation numerically. I was very interested in the possibility of approximating the solution.

Derivation

I will briefly recount what I tried, which follows the linked document closely.

Suppose I have an operator $A = A_0 + \epsilon A_1$ in dimensionless units, where $\epsilon \ll 1$. The equation I want to solve is $\dot{y} = A y$.

Suppose I assign two timescales, $t_0 = t$ and $t_1 = \epsilon t$, as described in the document. Then, when I take the time derivative and gather terms of the same power of $\epsilon$, I get

$$\left(\frac{\partial y_0}{\partial t_0} - A_0 y_0\right) + \epsilon \left(\frac{\partial y_0}{\partial t_1} + \frac{\partial y_1}{\partial t_0} - A_1 y_0 - A_0 y_1 \right) + \mathcal{O}\!\left(\epsilon^2\right) = 0$$

The solution to the first (order unity term) is

$$ y_0 = e^{A t_0} x_0\!\left(t_1\right) $$

and, when we solve the second term with this solution substituted,

$$ y_1 = e^{A_0 t_0} x_1\!\left(t_0\right) + e^{A_0 t_0} t_0 \left(A_1 x_0\!\left(t_1\right) - \frac{\partial x_0}{\partial t_1}\right) $$

Now, solving for the parenthetical term to go to zero (because it's secular),

$$x_0 = e^{A_1 t_1} z_0$$

Substituting this back into $y_0$ from earlier, we find that

$$y_0 = e^{A_0 t_0 + A_1 t_1} z_0 = e^{\left(A_0 + \epsilon A_1\right) t} z_0$$

which is exactly the solution to the linear equation.

Other attempts

I have tried to implement a kind-of IMEX (?) scheme, where I solve

$$ e^{\epsilon A_1 h} y^{n+1} = e^{A_0 h} y^n $$

which is implicit for the fast timescale and explicit for the slow timescale. This also doesn't work, because it is only valid until the simulation time reaches the slow timescale. The next logical thing to do is to have a transition from this scheme to a fully-implicit scheme when the time is large enough, but, in practice, that is too computationally expensive.

Question

I am curious to know if I've just done this wrong, but also if there are methods in the same spirit as the multiple-scale analysis that would work for a linear ODE.

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