Quantifying Resolutions for DNS and LES for Lax-Wendroff Method: Application to Uniform/Non-Uniform Compact Schemes

TL;DR

This paper uses global spectral analysis to evaluate and calibrate Lax-Wendroff methods with high-order compact schemes for DNS and LES of fluid flows, focusing on uniform and non-uniform grids.

Contribution

It introduces a GSA-based calibration of Lax-Wendroff methods with sixth-order compact schemes for non-uniform grids, applicable to DNS and LES of fluid flows.

Findings

GSA effectively calibrates dispersion relation preserving properties.

Non-uniform grid effects are quantified and compared to uniform grids.

The method shows promise for improving DNS and LES accuracy.

Abstract

The global spectral analysis (GSA) of numerical methods ensures that the dispersion relation preserving (DRP) property is calibrated in addition to ensuring numerical stability, as advocated in the von Neumann analysis. The DRP nature plays a major role where spatio-temporal dependence in the governing equation and boundary conditions has to be retained, such as in direct numerical simulations (DNS) and large eddy simulations (LES) of fluid flow transition. Using the concept of GSA, methods based on the Lax-Wendroff approach for temporal integration are calibrated using a high accuracy, sixth order non-uniform compact scheme, developed in "Hybrid sixth order spatial discretization scheme for non-uniform Cartesian grids - Sharma et al. Comput. Fluids, 157, 208-231 (2017)." The model equation used for this analysis is the one-dimensional (1D) convection-diffusion equation (CDE) which provides a unique state for the Lax-Wendroff method, results of which will have direct consequences for the solution of Navier-Stokes equations. Furthermore, the specific choice of the governing equation enables a direct assessment of the performance of numerical methods for solving fluid flows due to its one-to-one correspondence with the Navier-Stokes equation as established in "Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional Navier-Stokes equation - Suman et al. Comput. Fluids, 201, 104479 (2020)". The limiting case of the non-uniform compact scheme, which is a uniform grid, is considered. This is also investigated using GSA, and potential differences for the non-uniformity of grid are compared. Finally, further use of this newly developed Lax-Wendroff method for the non-uniformity of grid is quantified for its application in DNS and LES.