Preface

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This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all comamnds into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts to your needs for learning how to use the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

Runge--Kutta of order 2

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The second order Runge--Kutta method (denoted RK2) simulates the accuracy of the Tylor series method of order 2. Although this method is not as good as the RK4 method, its derivation illustrates all steps and the principles involved. We consider the initial value problem \( y' = f(x,y) , \quad y(x_0 ) = y_0 \) that is assumed to have a unique solution. Using a uniform grid of points \( x_n = x_0 + n\, h , \) with fixed step size h, we write down the Taylor series formula for y(x+h): where C is a constant expressed through the third derivative of y(x) and the other terms in the series involve powers of h greater than 3.

The derivatives y' and y'' must be extressed in terms of f(x,y) and its partial derivatives according to the given differential equation \( y' = f(x,y) . \) So we have

where \( f_x = \partial f/\partial x \quad \mbox{and} \quad f_y = \partial f/\partial y . \) Using these formulas, we derive the Taylor expression for y(x+h): Compare the above Runge-Kuta approximation of order 2 with the Taylor polynomial approximation we derive the following conclusions: Hence, if we require that coefficients in the RK2 method satisfy the relations Then the RK2 method will have the same order of accuracy as the Taylor's method, namely, 2.

Since there are only three equations in four unknowns, the system is underdetermined and we are permitted to choose one of the coefficients as we wish. There are several special choices that have been studied in the literature, and we mention some of them.

If we choose b₁ = 1/2, this yields b₂ = 1/2, c₂ = 1, and a_2,1 = 1. So we get Heun's method.

If we choose b₁ = 0, this yields b₂ = 1, c₂ = 1/2, and a_2,1 = 1/2. So we get the modified Euler method.

Example: The Butcher table for the explicit midpoint method is given by:

 

I. Second Order Optimal Runge-Kutta Approximation

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Ralston's method is a second-order method with two stages and a minimum local error bound. Its Mathematica realization is presented below when the step size is denoted by h: \begin{equation} \label{E35.opt} y_{n+1} =y_n + \frac{h}{4}\,f(x_n , y_n ) + \frac{3h}{4} \,f\left( x_n + \frac{2h}{3} , y_n + \frac{2h}{3} \,f(x_n , y_n ) \right) , \quad n=0,1,2,\ldots . \end{equation} Its Butcher tableau is

The same algorithm when the number of steps is specified:

If only one final value is needed, then Return command should be replaced with

 

II. Implicit Runge-Kutta of order 2

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\begin{equation} \label{E36.implicit} y_{n+1} = y_n + \frac{h}{2}\left( k_1 + k_2 \right) , \end{equation} where \begin{align*} k_1 &= f\left( x_n + \frac{\sqrt{3} -1}{2\sqrt{3}}\,h , \ y_n + \frac{h}{4}\,k_1 + \frac{\sqrt{3} -2}{4\sqrt{3}}\,hk_2\right) , \\ k_2 &= f\left( x_n + \frac{\sqrt{3} +1}{2\sqrt{3}}\,h , \ y_n + \frac{\sqrt{3} +2}{4\sqrt{3}}\,hk_1 + \frac{h}{4}\,k_2\right) , \end{align*}

First we present code when the step size (h) is given:

Then the same algorithm when the number of steps is specified:

Using Mathematica command NDSolve, we check the obtained value:

 

 

III. Ralston's method

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The method was proposed by Anthony Ralston (born in 1930, in New York City). Anthony Ralston received his Ph.D. from the Massachusetts Institute of Technology in 1956 under the direction of M. Eric Reissner. For approximating the solution of the initial value problem Ralston's second order method gives: where The Butcher tableau for Ralston's algorithm is

The second order Ralston approximation can written as follows:

\begin{equation*} %\label{E35.ralston} y_{n+1} =y_n + \frac{h}{3}\,f(x_n , y_n ) + \frac{2h}{3} \,f\left( x_n + \frac{3h}{4} , y_n + \frac{3h}{4} \,f(x_n , y_n ) \right) , \quad n=0,1,2,\ldots . \end{equation*}
The same algorithm with fixed step size (which is denoted by h) is specified:

Actually, the Heun, midpoint, and Ralston methods could be united into one algorithm, based on a parameter, which we denote by a2. Select a value for a2 between 0 and 1 according to
- Heun Method: a2 = 0.5
- Midpoint Method: a2 = 1.0
- Ralston's Method: a2 = 2/3

Then Mathematica codes will need only one modification:

Example: Consider the initial value problem for the Riccati equation

Mathematica is capable to find explicitly its solution (expressed through modified Bessel functions): To analize Runge--Kutta methods, we use Mathematica and type The following procedure estimates the solution of ordinary differential equations at a point xf.
n = number of steps
x0 = boundary condition for x
y0 = boundary condition for y
xf = value of x at which y is desired
f = differential equation in the form dy/dx = f(x,y)
a2 = constant to identify the numerical method

First, we determine the true numerical value at the final point:

Then we plot exact solution:

The following loop calculates the following quantiries:
AV = approximate value of the ODE solution usin
g second order RK method by calling the module RK2nd
Et = true error
et = absolute relative true error percentage
Ea = approximate error
ea = absolute relative approximate error percentage
sig = least number of significant digits correct in approximation

 

IV. Generic second-order method

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There is a family of Runge--Kutta methods of order 2, parameterized by an arbitrary positive number α from the interval (0,1] and given by the formula with Butcher tableau Correspondingly, where

This method unites all second-order Runge--Kutta methods; in particular, we have \( a= 1 \) for Heun's algorithm, \( a= 1/2 \) for the midpoint method, and \( a= 2/3 \) for optimal second order Ralston's algorithm.