Variable Time Step Method of DAHLQUIST, LINIGER and NEVANLINNA (DLN) for a Corrected Smagorinsky Model

TL;DR

This paper applies the DLN second-order, G-stable time-stepping method to a corrected Smagorinsky turbulence model, demonstrating unconditional stability, second-order convergence, and benefits of adaptive time stepping in turbulence simulations.

Contribution

It introduces the application and analysis of the DLN method for a corrected Smagorinsky model, including stability, convergence, and adaptive implementation insights.

Findings

Unconditionally stable for any time step sequence

Achieves second-order convergence in numerical solutions

Adaptive DLN controls numerical dissipation and reveals backscatter

Abstract

Turbulent flows strain resources, both memory and CPU speed. The DLN method has greater accuracy and allows larger time steps, requiring less memory and fewer FLOPS. The DLN method can also be implemented adaptively. The classical Smagorinsky model, as an effective way to approximate a (resolved) mean velocity, has recently been corrected to represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. In this paper, we apply a family of second-order, G-stable time-stepping methods proposed by Dahlquist, Liniger, and Nevanlinna (the DLN method) to one corrected Smagorinsky model and provide the detailed numerical analysis of the stability and consistency. We prove that the numerical solutions under any arbitrary time step sequences are unconditionally stable in the long term and converge at second order. We also provide error estimate under certain time step condition. Numerical tests are given to confirm the rate of convergence and also to show that the adaptive DLN algorithm helps to control numerical dissipation so that backscatter is visible.