On the Eigenvalues of the ADER-WENO Galerkin Predictor

TL;DR

This paper proves that the eigenvalues of matrices in the ADER-WENO Galerkin predictor are always zero across any dimensions and accuracy levels, ensuring desirable properties for high-order hyperbolic PDE solutions.

Contribution

It provides a rigorous proof that the eigenvalues of key matrices in ADER-WENO Galerkin predictors are always zero, regardless of dimensions or polynomial basis used.

Findings

Eigenvalues of matrices are always zero in the predictor system.

The result holds for any spatial dimension and order of accuracy.

The proof is basis-independent.

Abstract

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analytical derivation of the temporal derivatives of the solution required by the original ADER formulation has been replaced by the use of a cell-wise local Galerkin predictor.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.