The Montecinos-Balsara ADER-FV Polynomial Basis: Convergence Properties & Extension to Non-Conservative Multidimensional Systems

TL;DR

This paper extends the Montecinos-Balsara ADER-FV polynomial basis to multidimensional, non-conservative PDE systems, proving convergence properties and eigenvalue characteristics that enhance the method's efficiency for high-order hyperbolic PDE solutions.

Contribution

It introduces a new polynomial basis for multidimensional, non-conservative systems within the ADER-FV framework and proves key eigenvalue properties for this basis.

Findings

Eigenvalues of Galerkin matrices are always 0 for the new basis.

The new basis improves convergence efficiency in multidimensional systems.

Extension from conservative to non-conservative systems is successfully achieved.

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.