In trying to solve an ODE $y'=f(y,t)$ with a function f that is discontinuous at a subset (codim=1) of $\mathbb R^n$, I am looking for a Runge-Kutta ODE method whose stages do not evaluate $f(x,t)$ at $t+\Delta t$. So for example midpoint method:
Would be an acceptable 2nd order method.
Its Butcher block is
0 | 1/2 | 1 --------- 0 1
(If there's a nice way to typeset tables here, please let me know...)
the $c$ vector is the one consisting of 0 and 1/2 and as you see it is less than 1 in this case. I would like to find an explicit higher-order method that also has this property. All the higher-order (explicit) methods I have found seem to have at-least one element of $c$ equal to 1.
To summarize, my question is:
- Is there a known high-order (4 is enough) explicit Runge-Kutta method that does not evaluate the ODE at the end of the timestep?