Analysis of Stabilized Crank‐Nicolson Leapfrog Time‐Stepping Scheme for the Peterlin Viscoelastic Fluid Model

ABSTRACT In this paper, we propose a fully implicit and stabilized Crank‐Nicolson leapfrog (CNLF) time‐stepping scheme based on mixed finite element (MFE) spatial discretization for the non‐stationary Peterlin Viscoelastic Fluid model (PVFM). The innovation of this scheme lies in introducing a suitable stabilization term aimed at optimizing the structure and stability performance of the solution. By conducting meticulous mathematical derivations, we conclusively show that the proposed scheme possesses almost‐unconditional stability, provided that the time step remains confined within a specified constant range. In addition, employing the Stokes and Ritz projections, we present error estimates for essential physical quantities including velocity , pressure , and the conformation tensor , which maintain second‐order convergence accuracy in appropriate norms. To validate the practicality and effectiveness of this scheme, in the numerical test section, we initially present the linear CNLF numerical scheme. Two analysis numerical examples are provided: one for calculating the spatial convergence order and the other for determining the temporal convergence order. Additionally, we validate the stability of the scheme using driven cavity flow simulations, and compute the kinetic energy, entropy, and free energy.