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Solution Methods for IVPs: Runge-Kutta Methods

12.3.5.1 Runge-Kutta Methods

The Runge-Kutta methods developed by the German mathematicians C. Runge and M.W. Kutta are essentially a generalization of all the previous methods. Consider the following IVP:

    \[\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,t)\]


Assuming that the value of the dependent variable x (say x_i) is known at an initial value t_i, then, the Runge-Kutta methods employ the following general equation to calculate the value of x at t=t_{i+1}, namely x_{i+1} with h=t_{i+1}-t_i:

    \[x_{i+1}=x_i+h\phi\]


where \phi is a function of the form:

    \[\phi=\alpha_1 k_1+\alpha_2k_2+\alpha_3k_3+\cdots+\alpha_nk_n\]


\alpha_j are constants while k_j=F(x,t) evaluated at points within the interval [x_i,x_{i+1}] and have the form:

    \[\begin{split}k_1&=F(x_i,t_i)\\k_2&=F(x_i+q_{11}k_1h,t_i+p_1h)\\k_3&=F(x_i+q_{21}k_1h+q_{22}k_2h,t_i+p_2h)\\&\vdots\\k_n&=F(x_i+q_{(n-1),1}k_1h+q_{(n-1),2}k_2h+\cdots+q_{(n-1),(n-1)}k_{n-1}h,t_i+p_{n-1}h)\end{split}\]

The constants \alpha_j, and the forms of k_j are obtained by equating the value of x_{i+1} obtained using the Runge-Kutta equation to a particular form of the Taylor series. The k's are recurrence relationships, meaning k_1 appears in k_2, which appears in k_3, and so forth. This makes the method efficient for computer calculations. The error in a particular form depends on how many terms are used. The general forms of these Runge-Kutta methods could be implicit or explicit. For example, the explicit Euler method is a Runge-Kutta method with only one term with \alpha_1=1 and k_1=F(x_i,t_i). Heun’s method on the other hand is a Runge-Kutta method with the following non-zero terms:

    \[\begin{split}\alpha_1&=\alpha_2=\frac{1}{2}\\k_1&=F(x_i,t_i)\\k_2&=F(x_i+hk_1,t_i+h)\end{split}\]


Similarly, the midpoint method is a Runge-Kutta method with the following non-zero terms:

    \[\begin{split}\alpha_2&=1\\k_1&=F(x_i,t_i)\\k_2&=F(x_i+\frac{h}{2}k_1,t_i+\frac{h}{2})\end{split}\]


The most popular Runge-Kutta method is referred to as the “classical Runge-Kutta method”, or the “fourth-order Runge-Kutta method”. It has the following form:

    \[x_{i+1}=x_i+\frac{h}{6}\left(k_1+2k_2+2k_3+k_4\right)\]


with

    \[\begin{split}k_1&=F(x_i,t_i)\\k_2&=F\left(x_i+\frac{h}{2}k_1,t_i+\frac{h}{2}\right)\\k_3&=F\left(x_i+\frac{h}{2}k_2,t_i+\frac{h}{2}\right)\\k_4&=F(x_i+hk_3,t_i+h)\end{split}\]


The error in the fourth-order Runge-Kutta method is similar to that of the Simpson’s 1/3 rule, with a local error term E_{\mbox{local}}=\mathcal{O}(h^5) which would translate into a global error term of E=\mathcal{O}(h^4). The method is termed fourth-order because the error term is directly proportional to the step size raised to the power of 4.

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Example

Solve Example 4 above using the classical Runge-Kutta method but with h=0.4.

Solution

Recall the differential equation of this problem:

    \[\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,t)=tx^2+2x\]


Setting x_0=-5, t_0=0, h=0.4, the following are the values of k_1, k_2, k_3, and k_4 required to calculate x_1:

    \[\begin{split}k_1&=F(x_0,t_0)=F(-5,0)\\&=0+2(-5)=-10\\k_2&=F\left(x_0+\frac{h}{2}k_1,t_0+\frac{h}{2}\right)=F(-5+0.2(-10),0+0.2)\\&=0.2(-5+0.2(-10))^2+2\times (-5+0.2(-10))=-4.2\\k_3&=F\left(x_0+\frac{h}{2}k_2,t_0+\frac{h}{2}\right)=F(-5+0.2(-4.2),0+0.2)\\&=0.2(-5+0.2(-4.2))^2+2\times (-5+0.2(-4.2))=-4.8589\\k_4&=F(x_0+hk_3,t_0+h)=F(-5+0.4(-4.8589),0+0.4)\\&=0.4(-5+0.4(-4.8589))^2+2\times(-5+0.4(-4.8589))=5.3981\end{split}\]


Therefore:

    \[x_1=x_0+\frac{h}{6}(k_1+2k_2+2k_3+k_4)=-5+\frac{0.4}{6}(-10-2\times 4.2-2\times 4.8589+5.3981)=-6.51465\]

Proceeding iteratively gives the values of x_i up to t=5.2. The Microsoft Excel file RK4.xlsx provides the required calculations. The following Microsoft Excel table shows the required calculations:

RK42

The following Mathematica code implements the classical Runge-Kutta method for this problem with h=0.3. The output curve is shown underneath.

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RK41

MATLAB files: File 1 (ex12_10.m) File 2 (rk4.m)

The following tool provides a comparison between the explicit Euler method and the classical Runge-Kutta method. The classical Runge-Kutta method gives excellent predictions even when h=0.5, at which point the explicit Euler method fails to predict anything close to the exact equation. The classical Runge-Kutta method also provides better estimates than the midpoint method and Heun’s method above.




12.3.5.2 Lecture Video

 

 


 

 

 


 

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