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A decoupled Crank-Nicolson leap-frog scheme for the unsteady bioconvection flows problem with concentration dependent viscosity

Chenyang Li

TL;DR

This paper addresses unsteady bioconvection with concentration-dependent viscosity by formulating a coupled velocity–pressure–concentration system. It develops a decoupled Crank-Nicolson Leap-Frog (CNLF) finite element scheme using mini elements for $(\\mathbf{u},p)$ and a linear element for $c$, and proves unconditional stability via energy methods. Under Assumptions $A1$–$A2$ and a projection-based error framework, it establishes an $L^2$-error bound $\\max_{0\\le i \\le N-1} (\\|\\mathbf{u}^i-\\mathbf{u}_h^i\\|_{L^2}^2+\\|c^i-c_h^i\\|_{L^2}^2) \\le C(\\tau^4+h^4)$. Numerical experiments confirm second-order convergence in $L^2$ for $\\mathbf{u}$ and $c$, first-order for $p$, and robustness across viscosity models, highlighting improved efficiency over CN-based schemes.

Abstract

A fully discrete Crank--Nicolson Leap--Frog (CNLF) scheme is proposed and analyzed for the unsteady bioconvection flow problem with concentration-dependent viscosity. Spatial discretization is handled via the Galerkin finite element method (FEM), while temporal discretization employs the CNLF method for the linear terms and a semi-implicit approach for the nonlinear terms. The scheme is proven to be unconditionally stable, i.e., the time step is not subject to a restrictive upper bound. Using the energy method, $L^2$-optimal error estimates are derived for the velocity and concentration . Finally, numerical experiments are presented to validate the theoretical results.

A decoupled Crank-Nicolson leap-frog scheme for the unsteady bioconvection flows problem with concentration dependent viscosity

TL;DR

This paper addresses unsteady bioconvection with concentration-dependent viscosity by formulating a coupled velocity–pressure–concentration system. It develops a decoupled Crank-Nicolson Leap-Frog (CNLF) finite element scheme using mini elements for and a linear element for , and proves unconditional stability via energy methods. Under Assumptions – and a projection-based error framework, it establishes an -error bound . Numerical experiments confirm second-order convergence in for and , first-order for , and robustness across viscosity models, highlighting improved efficiency over CN-based schemes.

Abstract

A fully discrete Crank--Nicolson Leap--Frog (CNLF) scheme is proposed and analyzed for the unsteady bioconvection flow problem with concentration-dependent viscosity. Spatial discretization is handled via the Galerkin finite element method (FEM), while temporal discretization employs the CNLF method for the linear terms and a semi-implicit approach for the nonlinear terms. The scheme is proven to be unconditionally stable, i.e., the time step is not subject to a restrictive upper bound. Using the energy method, -optimal error estimates are derived for the velocity and concentration . Finally, numerical experiments are presented to validate the theoretical results.
Paper Structure (6 sections, 5 theorems, 72 equations, 3 figures, 12 tables)

This paper contains 6 sections, 5 theorems, 72 equations, 3 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and useful facts
  3. Decoupled Crank--Nicolson Leap-Frog FEM Algorithm and unconditional stability
  4. Convergent analysis
  5. Numerical experience
  6. Conclusion

Key Result

Lemma 2.1

(Discirete Gronwall's inequality ) Let $a_k , b_k$ and $y_k$ be the nonnegative numbers such that \newlabelbiobdf-11 Suppose $\tau \gamma _k \leq 1$ and set $\sigma_k = (1-\tau \gamma_k) ^{-1}$. Then there holds

Figures (3)

  • Figure 5.1: Convergence history of $(\mathbf{u},p,c)$ for $\nu=1$.
  • Figure 5.2: Convergence history of $(\mathbf{u},p,c)$ for $\nu=1+0.1c$.
  • Figure 5.3: Convergence history of $(\mathbf{u},p,c)$ for $\nu=exp(c)$.

Theorems & Definitions (10)

  • Lemma 2.1
  • Remark 2.1
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof