Divergence/connection preservation scheme in the curvilinear domain with a small geometric approximation error

TL;DR

This paper introduces two novel schemes to reduce geometric approximation errors in curvilinear domain discretizations, significantly improving conservation and long-term integration accuracy in numerical solutions of PDEs.

Contribution

The paper proposes two new schemes that recover the original divergence and connection in curvilinear domains, enhancing conservation and accuracy in numerical PDE solutions.

Findings

Improved conservation error performance demonstrated on PDEs with moving frames.

Schemes effectively reduce geometric approximation errors.

Enhanced long-time integration accuracy achieved.

Abstract

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.