solution of the diffusion equation by finite differences

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Solution of the Diffusion Equation by Finite

DifferencesThe basic idea of the finite differences method of solving PDEs is to replace spatial and time derivatives by

suitable approximations, then to numerically solve the resulting difference equations. Specifically, instead of

solving for with and continuous, we solve for , where

define the grid shown in Figure 1.

4/14/2011 Solution of the Diffusion Equation by Fi

www.me.ucsb.edu//node3.html 1


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Figure 1: Grid for our finite difference approximations. The point labelled corresponds

to , etc.

Derivatives of are approximated in terms of the values of at grid points. For example, we know that

This derivative evaluated at the grid point can be approximated in many different ways, the

simplest being the following:

Forward Difference:




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Backward Difference:

Central Difference:

The second derivative at the grid point may be approximated by using

Instead of using approximations for in terms of the values of at as for the forward difference,

or at the points as for the backward difference, let's imagine instead that we evaluate it at the

(fictitious) points , defined in the obvious way. Then, using central difference approximations for the

spatial derivatives evaluated at these points,

Then




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(3)

We can approximate derivatives with respect to time in the same way. For example, the forward difference

approximation for at the grid point is

(4)

It should be noted that these finite difference approximations are only valid to some order in or . The

error in the approximations is called the truncation error. It is possible to get approximations which are valid to

higher order by using more grid points in the approximations. This is all quite important, but for our purposes the

approximations given above will be sufficient.

Using the approximations (3) and (4) in (2), and rearranging, we get the following difference equation which can

be iterated to find the approximate solution to equation (2):

(5)

This is called an explicitnumerical scheme because the computation of at is completely determined by

our computation of at . This scheme is also called consistentbecause the finite difference approximations

have a truncation error that approaches zero in the limit that , .

Although this is a consistent method, we are still not guaranteed that iterating equation (5) will give a good

approximation to the true solution of the diffusion equation (2). A numerical scheme is called convergentif the

solution of the discretized equations (here, the solution of (5)) approaches the exact solution (here, the solution of

(2)) in the limit that , .

For linear equations such as the diffusion equation, the issue ofconvergence is intimately related to the issue of

stability of the numerical scheme (a scheme is called stable if it does not magnify errors that arise in the course

of the calculation). Indeed, the Lax Equivalence Theorem says that for a properly posed initial value problem for

a linear PDE, and a consistent finite difference approximation, stability is the necessary and sufficient condition fo




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convergence. Moreover, it can be shown that the scheme given by (5) is only convergent when

(6)

These issues are also quite important, but this is not the appropriate place to go into them in more detail. All of

the details are described, for example, inA First Course in Numerical Analysis of Differential Equations byArieh Iserles, Cambridge University Press, 1996.

However, before moving on, let me emphasize that as the sizes and are made smaller, the truncation

error of approximating the partial derivatives by finite differences decreases. However, for smaller sizes, more

computations need to be done to get solutions for the same domain and total time, which leads to increased

roundoff error. The total error as a function of these sizes is sketched in Figure 2.

Figure 2: Sketch of errors as functions of the grid size for a finite

difference calculation. Here it is assumed that the solution is

calculated on the same domain and for the same total time.

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Jeffrey M. Moehlis 2001-10-24



solution of the diffusion equation by finite differences

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