Algorithmic improvements for schemes using the ADER time discretization

We detail several algorithmic changes to the ADER Multi-Moment Finite-Volume Methods (ADER + MMFV) of Norman and Finkel (2012) [2]. The DT recurrence relations are improved, flux and source term differential transforms are saved, each are expanded as polynomials, quadrature is removed from the algorithm, integration is performed analytically, and a different Riemann solver admitting direct use of time-averaged fluxes is used. These algorithmic changes were implemented and tested, and smooth 1-D shallow water experiments confirm the same or slightly better accuracy as well as 2–3× lower runtimes compared to Norman and Finkel (2012) [2].