Haran Jackson

Haran Jackson

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analytical derivation of the temporal derivatives of the solution required by the original ADER formulation has been replaced by the use of a cell-wise local Galerkin predictor.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

Haran Jackson

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.

Haran Jackson

A new second-order numerical scheme based on an operator splitting is proposed for the Godunov-Peshkov-Romenski model of continuum mechanics. The homogeneous part of the system is solved with a finite volume method based on a WENO reconstruction, and the temporal ODEs are solved using some analytic results presented here. Whilst it is not possible to attain arbitrary-order accuracy with this scheme (as with ADER-WENO schemes used previously), the attainable order of accuracy is often sufficient, and solutions are computationally cheap when compared with other available schemes. The new scheme is compared with an ADER-WENO scheme for various test cases, and a convergence study is undertaken to demonstrate its order of accuracy.