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Suppose the first order differential equation

$\frac{dy}{dx}=f(x,y)$

If we look at a trivial finite difference lets say

$\frac{y^{n+1} - y^{n}}{\Delta t} = \phi f^{n+1} + (1-\phi)f^{n}$

We can observe that for different values of $\phi$, the error of the method will vary. For example we see if for $\phi=0,1,.5$ we see that the method simply reverts to forward euler, backward euler, and crank nickolson method.

This is my question, lets assume we didn't know what was the best value of $\phi$ was. Can we find the $\phi$ in some manner so we know that the local truncation error would be minimized. Maybe even more powerfully, find $\phi$, such that you find the $\phi$ which minimizes error, but is also stable( either unconditionally stable or a stability condition).

Thanks.

Vogtster
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  • I suspect this will depend on which function $f(x,y)$ you are using and at which $x$ and $y$ you are. – Kwin van der Veen Nov 21 '15 at 10:07
  • Use either $x$ or $t$ as independent variable. The method is the trapezoidal method, Crank-Nickolson is in 2 dimensions. See https://math.stackexchange.com/questions/1213614/numerical-estimates-for-the-convergence-order-of-trapezoidal-like-runge-kutta-me/1214079#1214079 for numerical experiments. – Lutz Lehmann Nov 21 '15 at 11:07

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