Title: Multidimensional upwind hydrodynamics on unstructured meshes using Graphics Processing Units I. Two-dimensional uniform meshes,
Abstract: We present a new method for numerical hydrodynamics which uses a
multidimensional generalisation of the Roe solver and operates on an
unstructured triangular mesh. The main advantage over traditional methods based
on Riemann solvers, which commonly use one-dimensional flux estimates as
building blocks for a multidimensional integration, is its inherently
multidimensional nature, and as a consequence its ability to recognise
multidimensional stationary states that are not hydrostatic. A second novelty
is the focus on Graphics Processing Units (GPUs). By tailoring the algorithms
specifically to GPUs we are able to get speedups of 100-250 compared to a
desktop machine. We compare the multidimensional upwind scheme to a
traditional, dimensionally split implementation of the Roe solver on several
test problems, and we find that the new method significantly outperforms the
Roe solver in almost all cases. This comes with increased computational costs
per time step, which makes the new method approximately a factor of 2 slower
than a dimensionally split scheme acting on a structured grid.