A linear-decoupled and unconditionally energy stable fully discrete scheme for Peterlin viscoelastic model

Abstract

In this paper, we design a linear-decoupled and unconditionally energy stable scheme utilizing the ZEC (“zero-energy-contribution”) technique for the diffusion Peterlin viscoelastic model. This model includes a diffusion term with an arbitrary small diffusion coefficient ε for the conformation tensor C. A specific ODE is introduced to deal with the nonlinear coupling terms for velocity u and C satisfying the ZEC property. We approximate the coupled nonlinear terms using the previous time-step results while still maintaining energy stability, allowing us to solve a linear-decoupled system at each time-step. Moreover, each component of C can be computed in parallel. We prove the unique solvability and energy stability of the fully discrete scheme. Additionally, we derive an error bound $C(\tau +h^2)$ for the P2/P1/P2 element, where the constant $C$ is not explicitly dependent on the reciprocal of ε. Several numerical experiments are presented to demonstrate the accuracy and performance of the proposed scheme. Comparison with a linear-decoupled scheme excluding the ZEC technique indicates that the proposed algorithm offers superior stability and performance.