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Hyperbolic systems of PDEs can be solved to arbitrary orders of accuracy byusing the ADER Finite Volume method. These PDE systems may be non-conservativeand non-homogeneous, and contain stiff source terms. ADER-FV requires aspatio-temporal polynomial reconstruction of the data in each spacetime cell,at each time step. This reconstruction is obtained as the root of a nonlinearsystem, resulting from the use of a Galerkin method. It was proved in Jackson[7] that for traditional choices of basis polynomials, the eigenvalues ofcertain matrices appearing in these nonlinear systems are always 0, regardlessof the number of spatial dimensions of the PDEs or the chosen order of accuracyof the ADER-FV method. This guarantees fast convergence to the Galerkin rootfor certain classes of PDEs.  In Montecinos and Balsara [9] a new, more efficient class of basispolynomials for the one-dimensional ADER-FV method was presented. This newclass of basis polynomials, originally presented for conservative systems, isextended to multidimensional, non-conservative systems here, and thecorresponding property regarding the eigenvalues of the Galerkin matrices isproved.

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