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.
翻译:当时间依赖部分差异方程式(PDEs)在具有曲线边界或曲线表面的域内以数字方式解决时,由于脊椎和其他内部网格点位置不准确而导致的网状错误和几何近似差差差差分别可能是PDE数字解决方案不准确和不稳定的主要原因。这些几何差错在破坏稳定性,特别是保护性能方面所起的作用基本上不为人知,这似乎需要非常精细的网格,特别是用来消除几何近似差差错。本文的目的是通过使用具有可忽略的几何近差差差误差的高阶网格差来调查几何近差差差差差的影响。为了实现这一目标,称为NekMesh的CAD几何方法的高端网格生成器,与传统的中观差差差差差差差差差,特别是用来消除几何近差差差差差差差差差差差差差差差。首先通过应用移动框架的方法,甚至可忽略几何几何近近近近差差差差差差差差差差差差差差差差差差差差差,然后通过在地理框架上采用移动方法,在数字精确度测算轨道上测量测得的精确度平差差差差区域内,对地差差差差差差差差差差差差差差差差差差差差差差差差差差差差差差差差差差差法,对地平平平平平平平平平平平平平比,在精确度上,对地平比,在精确度上,在精确度上,在精确度上,对地平差差差差差差差差差差差差差差差差差差差差差差差差差差差差的精确度上对差的精确度的精确度上对地平平平平比在精确度上对地平平比在精确度上对地平比在精确度上对地平比在精确度上研究。