Richardson extrapolation formula for Runge-Kutta method

Question

We use the 4-stage Runge Kutta method and we have computed $y_{n+1}^{(h)}$ and $y_{n+2}^{(h / 2)}$, two approximations of $y\left(t_{n}+h\right).$ Develop a formula for the Richardson extrapolation by assuming that the local truncation error is $C_{n} h^{5}$.

Answer 1

Starting from the same value $y_n$, performing $k$ steps with step size $h/k$ results in an approximation for the exact next value $y_{n+1}^*=y(t_{n+1};t_n,y_n)$ (exact solution to the IVP at $(t_n,y_n)$) with leading error term of the RK4 method according to $$ y^{(h/k)}_{n,k}=y_{n+1}^*+kC(t_n)(h/k)^5+O(h^6) $$ Comparing the results for $k=1$ and $k=2$ one gets approximations of the Richardson extrapolation $$ y_{n+1}^*=\frac{16y^{(h/2)}_{n,2}-y^{(h)}_{n,1}}{16-1}+O(h^6) $$ and the error estimate $$ y^{(h)}_{n,1}-y_{n+1}^*=\frac{16}{15}\left(y^{(h)}_{n,1}-y^{(h/2)}_{n,2}\right), $$ that is, at first the more precise value stands for the exact value, and then the difference is slightly corrected to get a better error estimate.