Title: The Montecinos-Balsara ADER-FV Polynomial Basis: Convergence Properties & Extension to Non-Conservative Multidimensional Systems,
Abstract: Hyperbolic systems of PDEs can be solved to arbitrary orders of accuracy by
using the ADER Finite Volume method. These PDE systems may be non-conservative
and non-homogeneous, and contain stiff source terms. ADER-FV requires a
spatio-temporal polynomial 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 was proved in Jackson
[7] that for traditional choices of basis polynomials, 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-FV method. This guarantees fast convergence to the Galerkin root
for certain classes of PDEs.
In Montecinos and Balsara [9] a new, more efficient class of basis
polynomials for the one-dimensional ADER-FV method was presented. This new
class of basis polynomials, originally presented for conservative systems, is
extended to multidimensional, non-conservative systems here, and the
corresponding property regarding the eigenvalues of the Galerkin matrices is
proved.