Riemann Problem for Linear Hyperbolic Systems
In this post, I will be solving the linear system

with the Riemann initial conditions

where and
is an n by n matrix which has distinct real eigenvalues.
Since has real distinct eigenvalues, we know that the matrix can be decomposed as

where columns in correspond to
‘s eigenvectors and
is a diagonal matrix containing the eigenvalues
![\displaystyle R = \left [ r_1 | r_2 | r_3|\cdots|r_n \right] \displaystyle R = \left [ r_1 | r_2 | r_3|\cdots|r_n \right]](http://jkwiens.com/2011/05/31/riemann-problem-for-linear-hyperbolic-systems/8b7d7c41010ad554487de113b168be01_3.png)

Additionally, because we have distinct real eigenvalues, we know that is a unitary matrix (
).
Because is unitary, we can uniquely decompose a vector into

Using this fact, it is beneficial to write and
as


where and
.
Now, we will transform the original pde using to


Likewise, we can rewrite the initial conditions as



where is a unit vector.
We can conclude that the original system decouples to n linear advection equations


which has the solution

Therefore, we can solve the original equation by using the transformation which gives us

