Scientific Computing

This module illustrates the numerical solution of the Lorenz system of ordinary differential equations, a crude model for atmospheric circulation. Ordinary differential equations (ODEs) have applications in a wide variety of contexts, including meteorology. One such application is modeling atmospheric circulation. In 1963 meteorologist Edward Lorenz published a model given by the system of ODEs

*y*_{1}′ = *σ* (*y*_{2} − *y*_{1}),

*y*_{2}′ = *r* *y*_{1} − *y*_{2} −
*y*_{1}* y*_{3},

*y*_{3}′ = *y*_{1} *y*_{2} − *b*
*y*_{3}.

This system of ODEs results from a spectral discretization of a partial
differential equation describing convective motion in a two-dimensional
fluid cell that is warmed from below and cooled from above, crudely
modeling atmospheric circulation. The variable
*y*_{1}*y*_{2}*y*_{3}*σ*
represents the Prandtl number, *r* the Rayleigh number, and
*b* the geometric proportions of the problem domain. This system
of ODEs was one of the first shown to exhibit *chaotic* behavior,
in which the solution oscillates in a seemingly random way, never
settling into either stationary nor truly periodic behavior.

The user first selects values for the parameters *σ*,
*r*, and *b*, whose default values are those used by Lorenz
in his original paper. Next the user selects the initial conditions
for each variable and how far forward in time to compute the solution.
Finally, the user selects the step size and numerical method to be used
in computing an approximate solution for the ODE system.

Clicking
*Solve* then calculates the approximate solution using the chosen
numerical method and step size, and displays the results graphically.
In the upper left graph, each component of the solution is plotted as a
function of time. The other three graphs plot the trajectories
*y*_{1}(*t*),
*y*_{2}(*t*)),*y*_{1}(*t*),
*y*_{3}(*t*))*y*_{2}(*t*),
*y*_{3}(*t*))

**References:**

- Michael T. Heath,
*Scientific Computing, An Introductory Survey*, 2nd edition, McGraw-Hill, New York, 2002. See Computer Problem 9.6 on page 419. - Edward N. Lorenz, Deterministic non-periodic flow,
*J. Atmos. Sci.*20:130-141, 1963.

**Developers:** Evan VanderZee and Michael Heath