How do test function analysis results carry over to general equations

numerical methods

In numerical analysis, when considering a numerical method to solve a differential equation (think of Euler Forward, Euler Backward, Runge-Kutta 4, etc.), usually all the basic properties of the method (truncation error, stability, convergence) are derived in the case the differential equation to be solved is $y'=\lambda y$.

I understand that investigating this equation can be insightful, because it is an easy one to solve analytically. However, what I don't get, is that all the said properties (truncation error, stability, convergence) are seemingly understood to hold in the general case (i.e., any ODE, not necessarily $y'=\lambda y$). How do test function analysis results carry over to general ODE's?

Best Answer

Basically the idea is that a general non-linear ODE can be linearized locally, given some reasonable constraints on the DE, and

Consider $y'(x) = f(y,x)$ with $y(x_0)=b$. Then consider: $$ y'(x) \approx f(b,x)+\frac{\partial f}{\partial y}\bigg\rvert_{y=b} [y-b]\approx \lambda y(x) + f(b,x) - \lambda b = \lambda y + c $$ where we have linearized about $y=b$, fixed some $x=x_0$, and let $\lambda=\partial_y f(b,x_0)$. Then of course we can solve the homogeneous ODE $y' =\lambda y$ to get a general solution locally.

The idea then is that if the algorithm has pleasant properties on this linearized version, then those properties carry over to the full ODE (as long as the ODE behaves nicely).

Here are some relevant links that go into more detail:

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