On the eigenvalues of the ADER-WENO Galerkin predictor

ADER-WENO methods represent an effective set of techniques for solving hyperbolic systems of PDEs. These systems may be non-conservative and non-homogeneous, and contain stiff source terms. The methods require a spatio-temporal reconstruction of the data in each spacetime cell, at each time step. This reconstruction is obtained as the root of a nonlinear system, resulting from the use of a Galerkin method. It is proved here that the eigenvalues of certain matrices appearing in these nonlinear systems are always 0, regardless of the number of spatial dimensions of the PDEs, or the chosen order of accuracy of the ADER-WENO method. This guarantees fast convergence to the Galerkin root for certain classes of PDEs.