High-order adaptive multi-resolution method on curvilinear grids I: Finite difference framework

Abstract

In this paper, we introduce a high-order adaptive multi-resolution method on curvilinear grids for solving hyperbolic conservation laws. To leverage the success of high-order and high-resolution schemes on Cartesian grids, the governing equations and physical variables on curvilinear grids are transformed into Cartesian grids, resulting in the introduction of additional terms related to geometric metrics. In order to achieve high-order accuracy, two techniques are employed to eliminate errors caused by geometric metrics and preserve the property of Geometric Conservation Laws (GCLs). Firstly, a newly developed and simple technique is applied to remove metric-related errors in the dissipation part of the WENO-ZQ scheme on curvilinear grids. Secondly, a GCLs-preserving data transfer operator is utilized to avoid errors caused by the metrics in the adaptive multi-resolution (MR) method on curvilinear grids. By combining these two techniques, a newly developed GCLs-preserving high-order adaptive multi-resolution method on curvilinear grids is obtained. The high-order accuracy, high resolution, and efficiency of the developed method are demonstrated through several benchmark tests conducted in one and two dimensions.