3.2 Linear hyperbolic systems of conservation laws

A, one-dimensional, linear hyperbolic system of partial differential equations (PDE) is

$$\frac{\partial {\bf u}}{\partial t} + {\bf A} \frac{\partial {\bf u}}{\partial x} = 0$$ (69)

where ${\bf A}$ is a constant $p\times p$ matrix with real eigenvalues $\lambda_i$ ($i=1\cdots p$)

By introducing the characteristic variables, ${\bf w}={\bf R}^{-1} {\bf u}$, system (70) can be rewritten

$$\frac{\partial {\bf w}}{\partial t} + {\bf\Lambda} \frac{\partial {\bf w}}{\partial x} = 0$$ (70)

where ${\bf\Lambda}={\bf R}^{-1}{\bf A}{\bf R}$, ${\bf\Lambda}=$diag( $\lambda_1 \cdots\lambda_p$) and ${\bf R}$ is the matrix of the right-eigenvectors (in columns)

Since ${\bf\Lambda}$ is diagonal, system (71) decouples into $p$ independent scalar equations

$$\frac{\partial w_i}{\partial t} + \lambda_i \frac{\partial w_i}{\partial x} = 0 i=1\cdots p$$ (71)

System (72) consists of constant coefficient linear advection equations, whose solution is

$$w_i(x,t)=w_i(x-\lambda_i t,0)$$ (72)

and for the original system  (70)

$${\bf u}(x,t) = {\bf R}{\bf w}(x,t)$$ (73)