7/19 Week
6:
Interpretation of Eigensystem
Different types of waves affect the state variables differently.
Each eigenvector corresponds to a different type of wave. The speed
of this wave is given by the eigenvalue. By finding the eigenvalues
and vectors we find the wave patterns. In the linear case, finding
the eigensystem allows the system of PDEs to be 'decoupled' into
individual scalar conservation laws. These equations can be solved
by tracing characteristics and resolving shockwaves.
We originally stated that all state variables
were functions of the two variables x and t. We now postulate that
there exists a class
of solutions, “Simple waves,” which depends on only one
parameter p. P is thus a real function of x and t. Simple waves ease
calculations, but also restrict the behavior of the system. Systems
with certain initial conditions cannot be solved in this way.
Because of simple waves, the Riemann Problem
takes on a new dimension of difficulty. Since not all initial profiles
can be resolved using
simple waves we must pick out those initial profiles that can be
resolved. We do this by setting the eigenvectors equal to the derivative
(with respect to p) of the vector of state variables and then solving
the resulting differential equations. Three vector functions of P
result.
U'(p) 
= 
First component of ith eigenvector 
V'(p) 
= 
Second component of ith eigenvector 
W'(p) 
= 
Third component of ith eigenvector 
