RK2#
- class brainpy.integrators.ode.RK2(f, beta=0.6666666666666666, var_type=None, dt=None, name=None, show_code=False, state_delays=None, neutral_delays=None)[source]#
Generic second order Runge-Kutta method for ODEs.
Derivation
In the
brainpy.integrators.ode.midpoint(),brainpy.integrators.ode.heun2(), andbrainpy.integrators.ode.ralston2(), we have already known first-order Euler methodbrainpy.integrators.ode.euler()has big estimation error.Here, we seek to derive a generic second order Runge-Kutta method [1] for the given ODE system with a given initial value,
\[y'(t) = f(t,y(t)), \qquad y(t_0)=y_0,\]we want to get a generic solution:
\[\begin{align} y_{n+1} &= y_{n} + h \left ( a_1 K_1 + a_2 K_2 \right ) \tag{1} \end{align}\]where \(a_1\) and \(a_2\) are some weights to be determined, and \(K_1\) and \(K_2\) are derivatives on the form:
\[\begin{align} K_1 & = f(t_n,y_n) \qquad \text{and} \qquad K_2 = f(t_n + p_1 h,y_n + p_2 K_1 h ) \tag{2} \end{align}\]By substitution of (2) in (1) we get:
\[\begin{align} y_{n+1} &= y_{n} + a_1 h f(t_n,y_n) + a_2 h f(t_n + p_1 h,y_n + p_2 K_1 h) \tag{3} \end{align}\]Now, we may find a Taylor-expansion of \(f(t_n + p_1 h, y_n + p_2 K_1 h )\)
\[\begin{split}\begin{align} f(t_n + p_1 h, y_n + p_2 K_1 h ) &= f + p_1 h f_t + p_2 K_1 h f_y + \text{h.o.t.} \nonumber \\ & = f + p_1 h f_t + p_2 h f f_y + \text{h.o.t.} \tag{4} \end{align}\end{split}\]where \(f_t \equiv \frac{\partial f}{\partial t}\) and \(f_y \equiv \frac{\partial f}{\partial y}\).
By substitution of (4) in (3) we eliminate the implicit dependency of \(y_{n+1}\)
\[\begin{split}\begin{align} y_{n+1} &= y_{n} + a_1 h f(t_n,y_n) + a_2 h \left (f + p_1 h f_t + p_2 h f f_y \right ) \nonumber \\ &= y_{n} + (a_1 + a_2) h f + \left (a_2 p_1 f_t + a_2 p_2 f f_y \right) h^2 \tag{5} \end{align}\end{split}\]In the next, we try to get the second order Taylor expansion of the solution:
\[\begin{align} y(t_n+h) = y_n + h y' + \frac{h^2}{2} y'' + O(h^3) \tag{6} \end{align}\]where the second order derivative is given by
\[\begin{align} y'' = \frac{d^2 y}{dt^2} = \frac{df}{dt} = \frac{\partial{f}}{\partial{t}} \frac{dt}{dt} + \frac{\partial{f}}{\partial{y}} \frac{dy}{dt} = f_t + f f_y \tag{7} \end{align}\]Substitution of (7) into (6) yields:
\[\begin{align} y(t_n+h) = y_n + h f + \frac{h^2}{2} \left (f_t + f f_y \right ) + O(h^3) \tag{8} \end{align}\]Finally, in order to approximate (8) by using (5), we get the generic second order Runge-Kutta method, where
\[\begin{split}\begin{aligned} a_1 + a_2 = 1 \\ a_2 p_1 = \frac{1}{2} \\ a_2 p_2 = \frac{1}{2}. \end{aligned}\end{split}\]Furthermore, let \(p_1=\beta\), we get
\[\begin{split}\begin{aligned} p_1 = & \beta \\ p_2 = & \beta \\ a_2 = &\frac{1}{2\beta} \\ a_1 = &1 - \frac{1}{2\beta} . \end{aligned}\end{split}\]Therefore, the corresponding Butcher tableau is:
\[\begin{split}\begin{array}{c|cc} 0 & 0 & 0 \\ \beta & \beta & 0 \\ \hline & 1 - {1 \over 2 * \beta} & {1 \over 2 * \beta} \end{array}\end{split}\]References