2.1 Characterization of the system of equations

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2.1 Characterization of the system of equations

has two distinct and real eigenvalues lambda₁<0<lambda₂,

such that we are dealing with a strictly hyperbolic system. We note that the celerity of gravitational waves c=sqrt(gh) takes the place the speed of sound has in gas dynamics and the Froude number Fr=|u|/c is the analogue to the Mach number.

Let us reduce the system to a diagonal form. For this we need the right eigenvectors rⁱ satisfying J-lambda_i*I)*rⁱ=0 which form the columns of the following matrix R,

with the inverse

The diagonalization of the jacobian J (denoting Lambda=diag(lambda₁,lambda₂)) can be written as:

In the linear case, i.e. when the Jacobian is constant, we have

where R^-1*dQ=dW defines the characteristic variables W_i which are advected along characteristic lines with respective wave speeds lambda_i.

2.1 Characterization of the system of equations

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