# Strong stability preserving RK scheme

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.

• You've seen "Heun's method", by any chance? – J. M. Jun 3 '13 at 18:31
• Thanks. Heun's method is what I was looking for. Similarly, is there an extension of the 3-stage SSPRK scheme ? – cfdlab Jun 4 '13 at 11:58
• 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". – Michael Jul 26 '16 at 15:02