Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes III. Code optimization via tensor contraction

Abstract

The present study deals with the code optimization and its implementation of the direct reconstruction method (DRM) using the complete-search tensor contraction (CsTC) framework to extract the best performance of high-order methods on modern computing architectures. DRM was originally proposed to overcome severe computational costs of the physical domain-based discontinuous Galerkin (DG) method on mixed-curved meshes. In this work, the performance of DRM is further enhanced through the code optimization via the CsTC technique. Required kernels for tensor operations in the DRM solution algorithm are analyzed and optimized by completely searching all candidates of GEMM (General Matrix Multiplication) subroutines. The computational performance is thoroughly examined by simulating a turbulent flow over a circular cylinder at ReD=3900 by DG-P3 and -P5 approximations. Compared to a quadrature-based approach with the full integration, the optimized DRM significantly reduces the memory requirements and the number of floating-point operations to compute the DG residual on a linear mesh as well as high-order curved meshes. On a P3-mesh, the optimized DRM provides 13.74× and 23.03× speed-ups in DG-P3 and -P5, respectively, while the amount of memory required is reduced to 1/16.6 and 1/19.9. On a linear mesh, it even yields 1.25× and 1.12× speed-ups in DG-P3 and -P5, respectively. The memory requirement is reduced to 1/1.27 and 1/1.15, respectively. In particular, it is observed that the optimized DRM on a P3-mesh performs better than the optimized quadrature-based method on a P1-mesh.