2019/08/23 by Sehun Chun, Julian Marcon, Chun, Sehun +6
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA
paper · pdf · doi:10.48550/arxiv.1908.09917
34 pages 22 figures
openalex publication_date 2019/08/23 · openalex created_date 2020/05/13 · arxiv created 2020/11/26 · arxiv updated 2020/11/30 · openalex updated_date 2026/07/28
When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. This paper aims to investigate the effect of geometric approximation error by using a high-order mesh with negligible geometric approximation error, even for high order polynomial of order p. To achieve this goal, the high-order mesh generator from CAD geometry called NekMesh is adapted for surface mesh generation in comparison to traditional meshes with non-negligible geometric approximation error. Two types of numerical tests are considered. Firstly, the accuracy of differential operators is compared for various p on a curved element of the sphere. Secondly, by applying the method of moving frames, four different time-dependent PDEs on the sphere are numerically solved to investigate the impact of geometric approximation error on the accuracy and conservation properties of high-order numerical schemes for PDEs on the sphere.