On the Eigenvalues of the ADER-WENO Galerkin Predictor
For researchers using ADER-WENO methods for hyperbolic PDEs, this proof ensures desirable system properties of the Galerkin predictor, removing a theoretical uncertainty.
This paper proves that the eigenvalues of certain matrices arising in the ADER-WENO Galerkin predictor are always zero for any spatial dimension, order of accuracy, and basis polynomial choice, confirming a previously unproven conjecture.
ADER-WENO methods have proved extremely useful in obtaining arbitrarily high-order solutions to problems involving hyperbolic systems of PDEs. For example, it has been demonstrated that for the same computational cost as a Runge-Kutta scheme of a certain order, one can obtain an ADER scheme of one higher order of accuracy. Additionally, Runge-Kutta schemes suffer from the presence of Butcher barriers, limiting the order of temporal accuracy that one can comfortably achieve. There are no such limitations present in ADER-WENO schemes. The cumbersome analytical derivation of the temporal derivatives of the solution required by the original ADER formulation has been replaced by the use of a cell-wise local Galerkin predictor. The predictor can take either a discontinuous or a continuous form. The Galerkin predictor is a high-order polynomial reconstruction of the data in both space and time, found as the root of a non-linear system. It has been conjectured that the eigenvalues of certain matrices appearing in these non-linear systems are always zero, leading to desirable system properties for certain classes of PDEs. It is proved here that this is in deed the case for any number of spatial dimensions and any desired order of accuracy, for both the discontinuous and continuous Galerkin variants. This result is independent of the choice of reconstruction basis polynomials.