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.

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We present first the Mean Value Theorem from calculus that have an impact on numerical methods. A special case of this theorem was first described by Parameshvara(1370--1460), from the Kerala school of astronomy and mathematics in India. A restricted form of the theorem was proved by the French mathematician Michel Rolle (1652--1719) in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by the French mathematician Baron Augustin-Louis Cauchy (1789--1857) in 1823.

Theorem (Mean Value Theorem): Let \( f:\, [a,b] \,\mapsto \,\mathbb{R} \) be a continuous function on the closed interval [a,b], and differentiable on the open interval (a,b), where a<b. Then there exists some ξ in (a,b) such that

\[ f(b) - f(a) = f' (\xi ) \left( b-a \right) . \]

Theorem (Cauchy Mean Value Theorem): If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval (a, b), then there exists some \( \xi \in (a,b) , \) such that

\[ \left( f(b) - f(a) \right) g' (\xi ) = f' (\xi ) \left( g(b) - g(a) \right) . \]

Theorem (Mean Value Theorem for definite integrals): If \( f:\, [a,b] \,\mapsto \,\mathbb{R} \) is integrable and \( g:\, [a,b] \,\mapsto \,\mathbb{R} \) is an integrable function that does not change sign on [a, b], then there exists some \( \xi \in (a,b) , \) such that

\[ \int_a^b g(t)\,f(t)\,{\text d} t = f(\xi ) \int_a^b g(t) \,{\text d} t . \]

Error Estimates

We discussed so far different numerical algorithms to solve the initial value problems for first order differential equations:

\[ y' = f\left( x, y \right) , \qquad y(x_0 )= y_0 , \]
where the slope function f is given and defined on some compact 2D domain to which the initial point \( \left( x_0 , y_0 \right) \) belongs. We associate with this problem a uniform grid (only for simplicity, in practice it is almost always variable) of points \( x_n = x_0 + n\,h , \quad n=0,1,2,,\ldots , \) where h is the step size. We always assume that the given initial value problem has a unique solution, which we denote by \( y= \phi (x) . \) A numerical approximation leads to the sequence of ordinates \( y_n , \quad n=0,1,2,\ldots , \) that we wish approximate the true values \( \phi (x_n ) . \) Their difference \( E_n = \phi (x_n ) -y_n \) is known as the global truncation error.

The methods we introduce for approximating the solution of an initial value problem are called difference methods or discrete variable methods. The solution is approximated at a set of discrete points called a grid (or mesh) of points. An elementary single-step method has the form \( y_{n=1} = y_n + h\, \Phi (x_n, y_n ) \) for some function Φ called an increment function.

The errors incurred in a single integration step are of two types: 1. The local truncation error or discretization error - the error introduced by the approximation of the differential equation by a difference equation. 2. Errors due to the deviation of the numerical solution from the exact theoretical solution of the difference equation. Included in this class are round-off errors, due to the inability of evaluating real numbers with infinite precision with the use of computer (i.e., computers usually operate on fixed word length) , and errors which are incurred if the difference equation is implicit and is not solved exactly at each step.
If a multistep method is used, an additional source of error results from the use of an auxiliary method (usually a single-step method, e.g., Runge--Kutta method), to develop the needed starting values for the multistep method.

In the numerical solution of an ODE, a sequence of approximations yn to the actual solution \( y = \phi (x ) \) is generated. Roughly speaking, the stability of a numerical method refers to the behavior of the difference or error, \( y_n - \phi (x_n ) , \) as n becomes large. To make our exposition simple, we consider the propagation of error in a one-step method by studying the particular representative one-step method of Euler:

\[ y_{n+1} = y_{n} + h\, f\left( x_n, y_n \right) , \quad n=0,1,2,\ldots . \]
This can be done by determining the relation of the error at step n+1 to the error at step n. To do this let \( \phi_n = \phi (x_n ) \) denote the true value at x = xn of the actual solution \( y = \phi (x) \) to the initial value problem \( y' = f(x,y) , \quad y(x_0 )= y_0 . \) For simplicity, we consider only the uniform grid of points \( x_n = x_0 + n\,h , \quad n=0,1,2,\ldots , \) with a fixed step size h. Then the total accumulated solution error εn at step n is defined by
\[ \varepsilon_n = \phi_n - y_n , \qquad n=0,1,2,\ldots . \]
The numerical values computed by Euler's algorithm satisfy the relation
\[ y_{n+1} = y_{n} + h\, f\left( x_n, y_n \right) - R_{n+1} , \]
where Rn+1, denotes the round-off errors resulting from evaluating Euler's rule. Similarly the true values of actual solution satisfy the relation
\[ \phi_{n+1} = \phi_{n} + h\, f\left( x_n, \phi_n \right) + T_{n+1} , \]
where Tn+1 denotes the local truncation error.

Subtracting the latter from the former yields

\[ \phi_{n+1} - y_{n+1} = \phi_{n} - y_n + h\left[ f\left( x_n, \phi_n \right) - f\left( x_n, y_n \right) \right] + E_{n+1} , \]
where \( E_{n+1} = T_{n+1} + R_{n+1} . \) This yields
\[ \varepsilon_{n+1} = \varepsilon_{n} + h\left[ f\left( x_n, \phi_n \right) - f\left( x_n, y_n \right) \right] + E_{n+1} . \]
Applying the mean value theorem to the terms inside the bracket, this relation between successive errors can be written as
\[ \varepsilon_{n+1} = \varepsilon_{n} + h\left( \phi_n - y_n \right) f_y \left( x_n, \eta_n \right) + E_{n+1} , \]
where fy denotes \( \frac{\partial f}{\partial y} \) and ηn lies between yn and \( \phi_n . \) If \( |f_y (x , y )| \le K \) and \( |E_{n+1} | \le E , \) where K and E are both positive constants, the error expression above can be replaced by a related first-order difference equation
\[ |\varepsilon_{n+1} | \le | \varepsilon_{n} | \left[ 1 + h\,K \right] + E . \]
Now if \( |\varepsilon_{n} | \le | e_n | , \) it follows that
\[ |\varepsilon_{n+1} | \le | e_{n} | \left[ 1 + h\,K \right] + E . \]
The propagated error is bounded by the solution of the related first-order difference equation
\[ e_{n+1} = e_{n} \, q + E ,\qquad \mbox{with the initial condition} \quad e_0 =0 , \]
where \( q = 1 + h\,K . \) This means that the error propagates linearly. If \( |q| = |1 + h\,K| > 1 , \) the error increases while for \( |q| = |1 + h\,K| < 1 , \) decreases. This observation leads to the condition:
\[ -2 < h\, K < 0 \qquad \mbox{or} \qquad \partial f/\partial y <0 . \]
The solution of this first-order difference equation for en can be found by successive substitution as follows since it is actually a geometric series:
\[ e_{n} = \frac{1- q^n}{1-q} \, E . \]
It follows then that the propagated error in Euler's algorithm is bounded by the expression
\[ |\varepsilon_{n+1} | \le \frac{\left( 1 + h\,K \right)^{n+1} -1}{h\,K} \, E . \]
Now we can make the following conclusions.


The global discretization error ek is defined by
\[ e_k = \phi (x_k ) - y_k , \qquad k=0,1,2,\ldots , m . \]
It is the difference between the unique solution and the solution obtained by the discrete variable method.

The local discretization error εk+1 is defined by

\[ \varepsilon_{k+1} = \phi (x_{k+1} ) - y_k - h\, \Phi (x_k , y_k ), \qquad k=0,1,2,\ldots , m-1 . \]
It is the error committed in the single step from xk to xk+1.

Theorem (Precision of Euler's Method): Assume that \( y= \phi (x) \) is the solution to the initial value problem

\[ y' = f(x,y) \qquad\mbox{over interval} \quad [a,b]=[x_0 , x_m ] \quad\mbox{subject to} \quad y(x_0 )= y_0 . \]
If \( \phi (x) \) has two continuous derivatives on the interval [a,b] and \( \left\{ \left( x_k , y_k \right) \right\}_{k=0}^m \) is the sequence of approximations generated by Euler's method, then
\[ \begin{split} |e_k | &= \left\vert \phi( x_k ) - y_k \right\vert = O(h) , \\ | \varepsilon_{k+1} | &= \left\vert \phi( x_{k+1} ) - y_k - h\, f(x_k , y_k ) \right\vert = O\left( h^2 \right) . \end{split} \]
The error at the end b = xm of teh interval [a,b] is called the final global error:
\[ E(\phi (b) , h) = \left\vert \phi (b) - y_m \right\vert = O(h) . \qquad ■ \]


Theorem (Precision of Heun's Method): Assume that \( y= \phi (x) \) is the solution to the initial value problem

\[ y' = f(x,y) \qquad\mbox{over interval} \quad [a,b]=[x_0 , x_m ] \quad\mbox{subject to} \quad y(x_0 )= y_0 . \]
If \( \phi (x) \) has three continuous derivatives on the interval [a,b] and \( \left\{ \left( x_k , y_k \right) \right\}_{k=0}^m \) is the sequence of approximations generated by Heun's method, then
\[ \begin{split} |e_k | &= \left\vert \phi( x_k ) - y_k \right\vert = O\left( h^2 \right) , \\ | \varepsilon_{k+1} | &= \left\vert \phi( x_{k+1} ) - y_k - h\, \Phi (x_k , y_k ) \right\vert = O\left( h^3 \right) , \end{split} \]
where \( \Phi (x_k , y_k ) = y_k + \left( \frac{h}{2} \right) \left[ f(x_k , y_k ) + f \left( x_{k+1} , y_k + h\,f(x_k ,y_k ) \right) \right] .\) The final global error at the end of the interval satisfies:
\[ E(\phi (b) , h) = \left\vert \phi (b) - y_m \right\vert = O\left( h^2 \right) . \qquad ■ \]


1.3.6. Accuracy

Sometimes you may be interested to find out what methods are being used in NDSolve.
Here you can see the coefficients of the default 2(1) embedded pair.

NDSolve`EmbeddedExplicitRungeKuttaCoefficients[2, Infinity]
{{{1}, {1/2, 1/2}}, {1/2, 1/2, 0}, {1, 1}, {-(1/2), 2/3, -(1/6)}}

You also may want to compare some of the different methods to see how they perform for a specific problem.

Implicit Runge--Kutta methods have a number of desirable properties.

The Gauss--Legendre methods, for example, are self-adjoint, meaning that they provide the same solution when integrating forward or backward in time.

A generic framework for implicit Runge--Kutta methods has been implemented. The focus so far is on methods with interesting geometric properties and currently covers the following schemes:


The derivation of the method coefficients can be carried out to arbitrary order and arbitrary precision.



Fixed Point Iteration

Bracketing Methods

Secant Methods

Euler's Methods

Heun Method

Runge-Kutta Methods

Runge-Kutta Methods of order 2

Runge-Kutta Methods of order 3

Runge-Kutta Methods of order 4

Polynomial Approximations

Error Estimates

Adomian Decomposition Method

Modified Decomposition Method

Multistep Methods

Multistep Methods of order 3

Multistep Methods of order 4

Milne Method

Hamming Method


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