Higher Order Gradients on Unstructured Meshes using Compact Formulation for Node-Centered Schemes

Paper in proceeding, 2022

This paper introduces a new approach for gradient computations on node-centered unstructured grids. The proposed approach is to derive a gradient algorithm from a least squares approximation, where a local system is solved to introduce connectivity between neighbouring nodes. The resulting scheme forms a globally coupled linear system of equations for the gradients which can be solved efficiently by iterative techniques. Fourth-order gradient accuracy for interior nodes can be obtained on isotropic quadrilateral and mixed elements (quadrilaterals and triangles) grids. The same accuracy is achieved on quadrilateral and mixed elements grids with high-aspect-ratio. Additionally, a different formulation to the standard distance weighted least squares node-centered gradients is proposed, which shows robust second-order accuracy on grids with high-aspect-ratio compared to the standard formulation. The paper concludes with a proposed continuation for future developments.