Two-level time-marching scheme using splines for solving the advection equation

A new numerical algorithm using quintic splines is developed and analyzed: quintic spline Taylor-series expansion (QSTSE). QSTSE is an Eulerian flux-based scheme that uses quintic splines to compute space derivatives and Taylor series expansion to march in time. The new scheme is strictly mass conservative and positive definite while maintaining high peak retention. The new algorithm is compared against accurate space derivatives (ASD), Galerkin finite element techniques, and the Bott scheme. The cases presented include classical rotational fields, deformative fields, as well as a full-scale aerosol model. Research shows that QSTSE presents significant improvements in speed and oscillation suppression against ASD. Furthermore, QSTSE predicts some of the most accurate results among the schemes tested.