Numerical Integration of Navier-Stokes Equations by Time Series Expansion and Stabilized FEM

TL;DR

This paper presents a novel numerical method combining Time Series Expansion and stabilized Finite Element Methods with Divergent Series Resummation for efficient and stable simulation of incompressible Navier-Stokes equations in fluid dynamics.

Contribution

It introduces a new stabilization strategy and series resummation technique within a FEM framework for improved Navier-Stokes equation integration.

Findings

Enhanced computational stability and accuracy in fluid flow simulations.

Successful application to laminar flow past a cylinder demonstrating method efficacy.

Significant improvement in series convergence and numerical robustness.

Abstract

This manuscript introduces an advanced numerical approach for the integration of incompressible Navier-Stokes (NS) equations using a Time Series Expansion (TSE) method within a Finite Element Method (FEM) framework. The technique is enhanced by a novel stabilization strategy, incorporating a Divergent Series Resummation (DSR) technique, which significantly augments the computational efficiency of the algorithm. The stabilization mechanism is meticulously designed to improve the stability and validity of computed series terms, enabling the application of the Factorial Series (FS) algorithm for series resummation. This approach is pivotal in addressing the challenges associated with the accurate and stable numerical solution of NS equations, which are critical in Computational Fluid Dynamics (CFD) applications. The manuscript elaborates on the variational formulation of Stokes problem and present convergence analysis of the method using the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. It is followed by the NS equations and the implementation details of the stabilization technique, underscored by numerical tests on laminar flow past a cylinder, showcasing the method's efficacy and potential for broad applicability in fluid dynamics simulations. The results of the stabilization indicate a substantial enhancement in computational stability and accuracy, offering a promising avenue for future research in the field.