Module

for

Parabolic P.D.E.'s

Background for Parabolic Equations

Heat Equation

As an example of
parabolic partial differential equations, we consider the one-dimensional heat equation

for   0 < x < a   and   0 < t < b.

with the initial condition

for    t = 0   and    .

and the boundary conditions

for   x = 0   and    ,

for   x = a   and    .

The heat equation models the temperature in an insulated rod with ends held at constant temperatures and and the initial temperature distribution along the rod being   f(x).  Although analytic solutions to the heat equation can be obtained with Fourier series, we use the problem as a prototype of a parabolic equation for numerical solution.

Computer Programs  Crank-Nicolson Method  Crank-Nicolson Method

Program (Forward-Difference method for the heat equation)  To approximate the solution of the heat equation    over the rectangle    with    ,  for  .  and  ,  for  .

``````

``````

Example 1.  Consider the heat equation where  .   The length of the rod is  .  Assume that the ends of the rod are held at the temperature  .  Assume that the initial temperature distribution is

.

Apply the forward difference method with    and  obtain temperature distributions for  .
We will use   .  This forces  .
Solution 1.

Example 2.  Consider the heat equation where  .   The length of the rod is  .  Assume that the ends of the rod are held at the temperature  .  Assume that the initial temperature distribution is

.

Now investigate what happens when the step size is too large.  This time the t interval is larger  [0.0, 0.35], and the step size is "too large."
Apply the forward difference method with    and  obtain temperature distributions for  .
We will use   .  This forces  .
Solution 2.

Crank-Nicolson Method

An implicit scheme, invented b
John Crank (1916-) and  Phyllis Nicolson (1917-1968), is based on numerical approximations for solutions at the point   that lies between the rows in the grid.  Specifically, the approximation used for    is obtained from the central-difference formula,

.

Computer Programs  Crank-Nicolson Method  Crank-Nicolson Method

Program (Crank-Nicolson method for the heat equation)  To approximate the solution of the heat equation    over the rectangle    with    ,  for  .  and  ,  for  .

``````

``````

Example 3.  Consider the heat equation where  .  The length of the rod is  .  Assume that the ends of the rod are held at the temperature  .  Assume that the initial temperature distribution is

.

Apply the Crank-Nicolson method with    and obtain temperature distributions for  .  Compare the solution with the exact solution:

.

We will use   .  This forces  This forces  .
Solution 3.

Example 4.  Consider the heat equation where  .  The length of the rod is  .  Assume that the ends of the rod are held at the temperature  .  Assume that the initial temperature distribution is

.

Apply the Crank-Nicolson method with    and obtain temperature distributions for  .  Compare the solution with the exact solution:

.

(Is the Crank-Nicolson method stable when r > 1 ?)
Solution 4.

Crank-Nicolson Method  Crank-Nicolson Method  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004