# Multidimensional upwind hydrodynamics on unstructured meshes using Graphics Processing Units I. Two-dimensional uniform meshes

## TL;DR

This paper introduces a multidimensional hydrodynamics method on unstructured meshes optimized for GPUs, outperforming traditional schemes in accuracy and recognizing complex stationary states, with significant speedups.

## Contribution

It presents a novel multidimensional Roe solver on unstructured meshes tailored for GPUs, enhancing accuracy and efficiency over traditional methods.

## Key findings

- Significantly outperforms traditional Roe solver in test problems
- Achieves 100-250x speedup on GPUs compared to desktop CPU
- Recognizes multidimensional stationary states beyond hydrostatic solutions

## 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.