Additional grid points are often introduced for the higher-order polynomial of a numerical solution with curvilinear elements. However, those points are likely to be located slightly outside the domain, even when the vertices of the curvilinear elements lie within the curved domain. This misallocation of grid points generates a mesh error, called geometric approximation error. This error is smaller than the discretization error but large enough to significantly degrade a long-time integration. Moreover, this mesh error is considered to be the leading cause of conservation error. Two novel schemes are proposed to improve conservation error and/or discretization error for long-time integration caused by geometric approximation error: The first scheme retrieves the original divergence of the original domain; the second scheme reconstructs the original path of differentiation, called connection, thus retrieving the original connection. The increased accuracies of the proposed schemes are demonstrated by the conservation error for various partial differential equations with moving frames on the sphere.
翻译:对于带有卷轴元素的数字解决方案的高阶多级多网点,通常会引入额外的网格点。但是,即使曲线线元素的脊椎位于曲线的域内,这些点也有可能位于域外的略微位置。这种网格点的错位会产生网状错误,称为几何近似差错。这个错误小于离散错误,但大到足以显著降解长期整合。此外,这一网格错误被认为是保存错误的主要原因。提出了两个新办法,以改善因几何近差差差错造成的长期整合中的保护错误和(或)离散错误:第一个办法检索原始域的原始差异;第二个办法重建了最初的区分路径,称为连接,从而重新定位原始连接。拟议方案增加的精度表现在区域上移动框架的各种部分差异方程式的保存错误中。