Reconstructed Multi-Moment Constrained Volume (RMCV) Schemes for Transport using ADER Time Discretizations

Modern computer architectures containing graphics processing units (GPUs) thrive in situations where a dense amount of computation can be performed on a set of data in an uninterrupted fashion (meaning no intermittent data traffic to and from Dynamic Random Access Memory, DRAM). Using these massively parallel architectures, we can consider much more computationally expensive time discretization techniques such as ADER DT (Arbitrary DERivatives in time and space with Differential Transforms) and limiting techniques like WENO ( Weighted Essentially Non-Oscillatory) limiting to improve accuracy of numerical schemes . We investigate the effect of techniques such as these on Mult-moment Constrained finite-Volume (MCV) schemes. MCV schemes are considered specifically for their ability to obtain high CFL values (particularly when paired with ADER time stepping). In particular, using ADER with third order MCV schemes achieve a maximum stable CFL=1 and fifth order schemes achieve a maximum stable CFL=.439 for one dimensional transport and dimensionally split two-dimensional transport. The third-order MCV scheme has a CFL value of 3 when calculated relative to the total number of Degrees of Freedom in an element, which is three times larger than regular Finite-Volume. For further reference, these CFL values are approximately four times larger than the CFL values of a nodal continuous Galerkin (i.e., "Spectral Element") scheme. In addition to these MCV schemes, we consider reconstruction based approaches for third-order MCV schemes utilizing neighboring information to maintain the large CFL value of the third-order MCV scheme. In addition to looking at the accuracy of these schemes and their ability to resolve features involving discontinuities, we perform computational performance analysis on OLCF's (Oak Ridge Leadership Computing Facility) Summit supercomputer comparing these schemes to other popular methods in simplified 2-D contexts.