The formulation of the schemes is based on piecewise linear functions in time that approximate displacements and momenta.
We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related details.
ADER schemes have been first conceived within the finite volume (FV) framework, but they were soon extended also to the discontinuous Galerkin (DG) finite element framework (Dumbser and Munz 2006; Taube et al. 2007) and to a unified formulation of FV and DG schemes, namely the so-called (mathbb{P}_{N}mathbb{P}_{M}) approach (Dumbser et al. 2008a).
This contributes (along with other intrinsic differences in the formulation of the schemes) to a significant speed-up of the code (up to ({sim}50%), depending on the setup of the grid on the computing nodes) with respect to MP5, since there is no need to compute the system eigenvectors and apply the resulting matrix.
An equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles.
The mathematical formulation of our scheme permits large scale realistic numerical simulations of busy intersections, allowing a more complete evaluation of the control performance, instead of just the collision avoidance at the intersection.
In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed.
The test platform developed in this paper has played an important role in the formulation of the system protection schemes of Liupanshui power grid (in Guizhou province).
However, the earlier formulations of the schemes are of central type (called central compact schemes, CCS), which are dispersive and susceptible to numerical instability.
In the present study, finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications.
