Numerical differential equation analysis package

Download .pdf, .docx, .epub, .txt
Did you like this example?

The Numerical Differential Equation Analysis package combines functionality for analyzing differential equations using Butcher trees, Gaussian quadrature, and Newton-Cotes quadrature.


Runge-Kutta methods are useful for numerically solving certain types of ordinary differential equations. Deriving high-order Runge-Kutta methods is no easy task, however. There are several reasons for this. The first difficulty is in finding the so-called order conditions. These are nonlinear equations in the coefficients for the method that must be satisfied to make the error in the method of order O (hn) for some integer n where h is the step size. The second difficulty is in solving these equations. Besides being nonlinear, there is generally no unique solution, and many heuristics and simplifying assumptions are usually made. Finally, there is the problem of combinatorial explosion. For a twelfth-order method there are 7813 order conditions!

Don’t waste time! Our writers will create an original "Numerical differential equation analysis package" essay for you whith a 15% discount.

Create order

This package performs the first task: finding the order conditions that must be satisfied. The result is expressed in terms of unknown coefficients aij, bj, and ci. The s-stage Runge-Kutta method to advance from x to x+h is then


Sums of the elements in the rows of the matrix [aij] occur repeatedly in the conditions imposed on aij and bj. In recognition of this and as a notational convenience it is usual to introduce the coefficients ci and the definition

This definition is referred to as the row-sum condition and is the first in a sequence of row-simplifying conditions.

If aij=0 for all i≤j the method is explicit; that is, each of the Yi (x+h) is defined in terms of previously computed values. If the matrix [aij] is not strictly lower triangular, the method is implicit and requires the solution of a (generally nonlinear) system of equations for each timestep. A diagonally implicit method has aij=0 for all i<j.

There are several ways to express the order conditions. If the number of stages s is specified as a positive integer, the order conditions are expressed in terms of sums of explicit terms. If the number of stages is specified as a symbol, the order conditions will involve symbolic sums. If the number of stages is not specified at all, the order conditions will be expressed in stage-independent tensor notation. In addition to the matrix a and the vectors b and c, this notation involves the vector e, which is composed of all ones. This notation has two distinct advantages: it is independent of the number of stages s and it is independent of the particular Runge-Kutta method.

For further details of the theory see the references.


the coefficient of f(Yj(x)) in the formula for Yi(x) of the method


the coefficient of f(Yj(x)) in the formula for Y(x) of the method


a notational convenience for aij


a notational convenience for the vector (1,

Do you want to see the Full Version?

View full version

Having doubts about how to write your paper correctly?

Our editors will help you fix any mistakes and get an A+!

Get started
Leave your email and we will send a sample to you.
Thank you!

We will send an essay sample to you in 2 Hours. If you need help faster you can always use our custom writing service.

Get help with my paper
Sorry, but copying text is forbidden on this website. You can leave an email and we will send it to you.