MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Return to Question

2 deleted 348 characters in body

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$. So for example midpoint method:$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$vector is the one consisting of 0 and 1/2 and as you see it is less than 1 in this case. I would like to find a an explicit higher-order method that also has this property, however, all . All the higher-order (explicit) methods I have found seem to have at-least one element of$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]$(with similarly high accuracy). To summarize, my questions arequestion is: 1. Is there a known high-order (4 is enough) explicit Runge-Kutta method that does not evaluate the ODE at the end of the timestep? 2. Is there way a way to evaluate the solution at arbitrary points inside the timestep, after obtaining the stages? I am aware of the theory behind RK methods and could task myself with deriving such a method, but if it has been done, I'm sure it will be better than what I'll come up with.... 1 # Runge-Kutta method with c<1 In trying to solve an especially stiff ODE$y'=f(y,t)$, I am looking for a Runge-Kutta ODE method whose stages do not evaluate$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$vector is the one consisting of 0 and 1/2 and as you see it is less than 1. I would like to find a higher-order method that also has this property, however, all the higher-order methods I found seem to have at-least one element of$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 are:

1. Is there a known high-order Runge-Kutta method that does not evaluate the ODE at the end of the timestep?
2. 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....