On the GPU performance of cell-centered finite volume method over unstructured tetrahedral meshes

Finite volume methods are widely used numerical strategies for solving partial differential equations. This paper aims at obtaining a quantitative understanding of the achievable GPU performance of finite volume computations in the context of the cell-centered finite volume method on 3D unstructured tetrahedral meshes. By using an optimized implementation and a synthetic connectivity matrix that exhibits a perfect structure of equal-sized blocks lying on the main diagonal, we can closely relate the achievable computing performance to the size of these diagonal blocks. Moreover, we have derived a theoretical model for identifying characteristic levels of the attainable performance as function of the GPU's key hardware parameters. A realistic upper limit of the performance can thus be accurately predicted. For real-world tetrahedral meshes, the key to high performance lies in a reordering of the tetrahedra, such that the resulting connectivity matrix resembles a block diagonal form where the optimal size of the blocks depends on the GPU hardware. Performance can then be predicted accurately based on the success of the reordering. Numerical experiments confirm that the achieved performance is close to the practically attainable maximum and it reaches 75% of the theoretical upper limit, independent of the actual tetrahedral mesh considered.