In trying to solve an especially stiff ODE $y'=f(y,t)$, 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 a an explicit higher-order method that also has this property, however, all . All the higher-order (explicit) methods I have found seem to have at-least one element of $c$ equal to 1(and claims that this is the most efficient). In addition I would like to be able to evaluate the solution for any $t'\in[t,t+\Delta t]$ (with similarly high accuracy).
To summarize, my questions arequestion 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?
- Is there way a way to evaluate the solution at arbitrary points inside the timestep, after obtaining the stages?
I am aware of the theory behind RK methods and could task myself with deriving such a method, but if it has been done, I'm sure it will be better than what I'll come up with....