A positive-definite, WENO-limited, high-order finite volume solver for 2-D transport on the cubed sphere using an ADER time discretization

Modern computer architectures reward added computation if it reduces algorithmic dependence, reduces data movement, increases accuracy/robustness, and improves memory accesses. The driving motive for this study is to develop a numerical algorithm that respects these constraints while improving accuracy and robustness. This study introduces the ADER-DT (Arbitrary DERivatives in time and space-differential transform) time discretization to positive-definite, weighted essentially nonoscillatory (WENO)-limited, finite volume transport on the cubed sphere in lieu of semidiscrete integrators. The cost of the ADER-DT algorithm is significantly improved from previous implementations without affecting accuracy. A new function-based WENO implementation is also detailed for use with the ADER-DT time discretization. While ADER-DT costs about 1.5 times more than a fourth-order, five-stage strong stability preserving Runge-Kutta (SSPRK4) method, it is far more computationally dense (which is advantageous on accelerators such as graphics processing units), and it has a larger effective maximum stable time step. ADER-DT errors converge more quickly with grid refinement than SSPRK4, giving 6.5 times less error in the L-infinity norm than SSPRK4 at the highest refinement level for smooth data. For nonsmooth data, ADER-DT resolves C-0 discontinuities more sharply. For a complex flow field, ADER exhibits less phase error than SSPRK4. Improving both accuracy and robustness as well as better respecting modern computational efficiency requirements, we believe the method presented herein is competitive for efficiently transporting tracers over the sphere for applications targeting modern computing architectures.