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
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