Determining Parameter Ranges for High Accuracy Large Eddy Simulation by Lax-Wendroff Method

TL;DR

This paper analyzes the Lax-Wendroff method for high-accuracy large eddy simulation of convection-diffusion equations using spectral analysis, establishing parameter limits and validating with lid-driven cavity flow simulations.

Contribution

It develops a spectral plane analysis for the Lax-Wendroff method applied to convection-diffusion equations, providing parameter calibration guidelines for high-accuracy simulations.

Findings

Derived limits for numerical parameters of LW method

Mapped numerical properties for LES applications

Validated with lid-driven cavity flow simulations at high Reynolds number

Abstract

The analysis of Lax-Wendroff (LW) method is performed by the generic modified differential equation (MDE) approach in the spectral plane using Fourier transform. In this approach, the concept of dispersion relation plays a major role relating spatial and temporal dependence of the governing differential equation, including initial and boundary conditions in developing high accuracy schemes. Such dispersion relation preserving schemes are calibrated in the spectral plane using the global spectral analysis for the numerical method in the full domain. In this framework, the numerical methods are calibrated by studying convection and diffusion as the underlying physical processes for this canonical model problem. In the LW method spatial and temporal discretizations are considered together, with time derivatives replaced by corresponding spatial derivatives using the governing equation. Here the LW method is studied for the convection-diffusion equation (CDE) to establish limits for numerical parameters for an explicit central difference scheme that invokes third and fourth spatial derivatives in the MDE, in its general form. Thus, for the LW method, two different MDEs are obtained, depending on whether the LW method is applied only on the convection operator, or both on the convection and diffusion operators. Motivated by a one-to-one correspondence of the Navier-Stokes equation with the linear CDE 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)", an assessment is made here to solve flow problems by these two variants of the LW method. Apart from mapping the numerical properties for performing large eddy simulation for the LW methods, simulations of the canonical lid-driven cavity problem are performed for a super-critical Reynolds number for a uniform grid.