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:

$k_1=f(x,t)$

$k_2=f(x+k_1/2,t+\Delta t/2)$

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?