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.