Date: 10.2.1999
Janne Viljas, 44468K

Session 2, small projects:
The method of characteristics

This page is still a bit unfinished. Unfortunately the subject is quite analytical and very little numerics was considered in the litterature that was available to me. Hence, very litlle numerical things will be presented here also. In fact, I think I just about covered everything essential in my presentation.

Analytic examples

1. A simple example

Let's use the ideas of the method of characteristics to solve the equation 
exp(t) f_x(x,t) + f_t(x,t) = 0  for t > 0
I think I will use Maple to make it look cool.

2. Other simple examples

I'll show more examples on the blackboard or as viewgraphs. At least one containing a nonhomogeneous equation (nonzero c in the equation (2)), or one with a term linear in the solution function f itself (nonzero d). In such cases f is no longer constant along the characteristics, but still solves an ODE other than df/dp = 0. If I failed to stress it before, I'll do it now: the whole point of the method of characteristics is to reduce a PDE to an ODE along some curves, which are then called the characteristics of the PDE. An important example would also be one which shows that a point in the xt-plane cannot always be traced back to the initial value curve, like some point on the x-axis. In other words the relation x_0=p(x,y) does not exist for all (x,t). Hence, the given initial data only defines the solution in part of the xt-plane!

Matlab stuff

Here I will try to present some solutions of problems in Chapter 2 of Cooper's book, which require the use of Matlab.

Snapshots

These have to do with exercises 5, 6 and 7.

Exercise 5 a This image shows solution curves for u_t + x u_x = 0 at selected timesteps when the inital curve on the x-axis is gaussian. The spreading of the characteristics is seen as a spreading of the solution!

Exercise 5 b This image was plotted to examine the speed of propagation of the top of the wave. Plots were made at times t=0.5,0.5+dt and t=1.5,1.5+dt with dt=0.1. Using the zoom feature of Matlab one can approximate the speeds directly from the image.

Exercise 7 c This image shows solutions at some equally spaced timesteps for the equation 

u_t + (1/x) u_x = 0  on x > 0    u(x,0) = f(x)
The characteristics are now parabolas and the solutions do not spread. (At least they shouldn't...) But this is a case where the solution is only defined for (x,t) which are to the right of the parabola going through the origin, x^2 > 2t. For points that do not satisfy this, the corresponding parabola is completely above the x-axis and consequently never gets a chance to pick up an initial value from it.

Shocks in solutions

This image was plotted with the mtc.m script, which solves nonlinear equations of the form u_t + u u_x = 0, u(x,0) = f(x) using the method of charcteristics. Profile 1 was chosen, with time steps 0,1,2,3 and 4. A shock (discontinuity) should develop between t=2 and t=3 in order for the solution to define a function.

Here is another image, created with the shock.m script. It does the same thing as mtc.m, but it develops the shock as it should. Time steps 0,2,3,4 and 5 for profile 1 are shown. The shock is due to a crossover of characteristics.

Maple PDEtools: PDEplot

Here's a link to the HTML code exported from Maple. It's a simple example of the use of PDEplot in the (P)DEtools package. This is the same thing as a Maple worksheet. This illustrates a similar situation as in the Matlab's mtc.m-case, where a shock should develop when characteristics cross.

(By the way, I hate it when xmaple starts to give segmentation faults! Another reason for xmaple to crash has been an illegal instruction. This is NOT a very stable program, is it?)