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For the ODE

$$ \dot{x} = f(x) $$

we have the 2-stage, second order SSP RK scheme (Shu, Osher, Gottlieb)

$$ x^{(0)} = x^n $$

$$ x^{(1)} = x^{(0)} + \Delta t f(x^{(0)}) $$

$$ x^{(2)} = \frac{1}{2} x^n + \frac{1}{2}[ x^{(1)} + \Delta t f(x^{(1)})] $$

$$ x^{n+1} = x^{(2)} $$

What is the scheme in case of the following ODE

$$ \dot{x} = f(x,t) $$

In particular I am looking for papers which deal with this case.

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  • $\begingroup$ You've seen "Heun's method", by any chance? $\endgroup$ – J. M. Jun 3 '13 at 18:31
  • $\begingroup$ Thanks. Heun's method is what I was looking for. Similarly, is there an extension of the 3-stage SSPRK scheme ? $\endgroup$ – cfdlab Jun 4 '13 at 11:58
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    $\begingroup$ The operator $f$ could also depend explicitly on $t$. See Equation (2.11) in Osher and Shu' paper "Efficient implementation of essentially non-oscillatory shock-capturing schemes". $\endgroup$ – Michael Jul 26 '16 at 15:02

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