ODEs 2

• Write down the expression for the modified midpoint (MM) method
• Write down the expression for the modified Euler (ME) method
  • How is this different from the implicit trapazoid (IT) method?
• What is the global order of the MM, ME, and IT methods? Write this in big-O notation.
• How many second order Runge Kutta methods are there?
• For the ODE $dy/dt = f(y)$, ignoring the time dependence here, the second-order Runge-Kutta method has the form $$y_{k+1} = y_k + h(c_1S_1 + c_2S_2),$$ $$S_1 = f(y_k),$$ $$S_2 = f(y_k+\beta hS_1).$$
  • What is the constraint on the values of $c$?
• Write down the formula for the fourth order RK method
• Why are methods $\le$ forth order special in terms of function evaluations?