Second-order Runge-Kutta scheme (RK2)
There are several variations of the second order Runge-Kutta schemes of numerical integration.
They are predictor-correction methods that it involves 2 stages. A step that predicts the value at the next time step (using the explicit Euler's method as predictor) and another step that applies a correction to the estimated value.
Put simply, it improves upon the Euler method by adding a midpoint in the step which increases the accuracy by one order.
The local error at each step is order \(O(h^3)\) giving a global error of order \(O(h^2)\).
Heun method
The Heun's method is expressed as follows:
\[y_{n+1} = y_n + \frac{h}{2}[f(t_n, y_n) + f(t_{n+1},y_{n+1})]\]
where:
- \(y_n\) is the approximation of solution at time \(t_n\)
- \(y_{n+1}\) is the approximation of solution at time \(t_{n+1}\)
- \(h\) is the step size
- \(f(t,y)\) is the function that defines the derivative of \(y\) with repsect to \(t\), ie \(\dot y = f(t, y)\)
The two steps are as follows:
- In the first step, calculate the estimate of slope of solution at beginning of time step. Then calculate a provisional value of the solution at the end of the time step, by using explicit Euler.
- Calculate the estimate of slope of solution at provisional value of the solution. Then use this estimate to calculate a new improved estimate of solution at end of time step.
To make the above single line formula easier to understand, the method may be expressed as follows:
\[y_{n+1}=h\frac{1}{2}\left[k_1+k_2 \right]\]
where:
\begin{aligned} k_{1}&=\ f(t_{n},y_{n}),\\ k_{2}&=\ f(t_{n+1},hf(t_{n+1}, y_{n+1})) = f(t_{n+1},hk_1) \end{aligned}
\[\begin{aligned}p_1&=x_n\\v_1&=v_n\\a_1&=a(p_1,v_1)\\p_2&=p_1+v_1\cdot\Delta t\\v_2&=v_1+a_1\cdot \Delta t\\a_2&=a(p_2,v_2)\\x_{n+1}&=x_n+\frac{v_1+v_2}{2}\cdot \Delta t\\v_{n+1}&=v_n+\frac{a_1+a_2}{2}\cdot \Delta t \end{aligned} \]
Comparison to basic Euler
In essence, where as the basic forward Euler method takes the slope of the solution curve at the beginning of the timestep \((k_1)\) and extrapolates the solution at the end of the time step \(t_n\).
The Heun method estimates a better slope by taking the average of the slope at the beginning of the time step and the end of the timestep. However we do not know the slope at \(t_{n+1}\). So we estimate it using basic Euler.
Midpoint method
Explicit Midpoint method
The explicit midpoint method, sometimes known as the modified Euler method is:
\[y_{n+1}=y_n + hf(t_n+h/2, y_n+hf(t_n,y_n))\]
or,
\[y_{n+1}=y_n+hk_2\]
where:
\[\begin{aligned} k_{1}&=f(t_{n},y_{n}), \\ k_{2}&=f(t_{n+\frac{1}{2}},y_{n+\frac{1}{2}}) = f(t_{n+\frac{1}{2}},hk_1) \end{aligned}\]
Implicit Midpoint method
\[y_{n+1}=y_n + hf(t_{n+\frac{h}{2}}, \frac{1}{2}(y_n+y_{n+1}))\]
Fourth-order Runge Kutta method (RK4)
The "classic Runge-Kutta method" or simply the "Runge-Kutta method" refers to the following.
The formula is given by
\[\begin{aligned} y_{n+1}&=y_{n}+{\frac {h}{6}}\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right),\\ t_{n+1}&=t_{n}+h\\ \end{aligned} \]
for n = 0, 1, 2, 3, ..., using:
\[{\displaystyle {\begin{aligned}k_{1}&=\ f(t_{n},y_{n}),\\k_{2}&=\ f\!\left(t_{n}+{\frac {h}{2}},y_{n}+h{\frac {k_{1}}{2}}\right),\\k_{3}&=\ f\!\left(t_{n}+{\frac {h}{2}},y_{n}+h{\frac {k_{2}}{2}}\right),\\k_{4}&=\ f\!\left(t_{n}+h,y_{n}+hk_{3}\right)\end{aligned}}}\]
Here \(y_{n+1}\) is the RK4 approximation of \(y(t_{n+1})\), and the next value \((y_{n+1})\) is determined by the present value \((y_{n})\) plus the weighted average of four increments, where each increment is the product of the size of the interval, h, and an estimated slope specified by function f on the right-hand side of the differential equation.
