Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameter Error Analysis for the 3D Modified Leray-alpha Model: Analytical and Numerical Approaches

Débora A. F. Albanez, Maicon J. Benvenutti, Samuel Little, Jing Tian

Abstract

In this study, we conduct a parameter error analysis for the 3D modified Leray-$α$ model using both analytical and numerical approaches. We first prove the global well-posedness and continuous dependence of initial data for the assimilated system. Furthermore, given sufficient conditions on the physical parameters and norms of the true solution, we demonstrate that the true solution can be recovered from the approximation solution, with an error determined by the discrepancy between the true and approximating parameters. Numerical simulations are provided to validate the convergence criteria.

Parameter Error Analysis for the 3D Modified Leray-alpha Model: Analytical and Numerical Approaches

Abstract

In this study, we conduct a parameter error analysis for the 3D modified Leray- model using both analytical and numerical approaches. We first prove the global well-posedness and continuous dependence of initial data for the assimilated system. Furthermore, given sufficient conditions on the physical parameters and norms of the true solution, we demonstrate that the true solution can be recovered from the approximation solution, with an error determined by the discrepancy between the true and approximating parameters. Numerical simulations are provided to validate the convergence criteria.

Paper Structure

This paper contains 12 sections, 6 theorems, 92 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions and inequalities
  4. Estimates for the ML-$\alpha$ solutions
  5. Regularity Analysis and Error Estimates
  6. Regularity analysis for the data assimilation system
  7. Long-time error analysis
  8. Numerical Simulations
  9. Testing the impact of $\eta$-without random initial conditions
  10. Testing the impact of $\eta$-with random initial conditions
  11. Discussions on the numerical computations
  12. Conclusions

Key Result

Theorem 1

Let $f\in L^{2}([0,T];H)$, $u(0)=u_{0}\in V$, and $T>0$, the system MLalpha has a unique regular solution $u\in C([0,T);V)\,\cap\, L^{2}([0,T];D(A))$, with $\frac{du}{dt}\in L^{2}([0,T];H)$.

Figures (12)

  • Figure 1: Error plot of modified Leray-$\alpha$ model with high $\eta$ value-without random initial conditions case.
  • Figure 2: Velocity contour of modified Leray-$\alpha$ model with high $\eta$ value-without random initial conditions case at $t=0$.
  • Figure 3: Velocity contour of modified Leray-$\alpha$ model with high $\eta$ value-without random initial conditions case at $t=400$.
  • Figure 4: Error plot of modified Leray-$\alpha$ model with low $\eta$ value-without random initial conditions case.
  • Figure 5: Velocity contour of modified Leray-$\alpha$ model with low $\eta$ value-without random initial conditions case at $t=0$.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Theorem 1: Existence and Uniqueness of Regular Solutions ilyin2006modified
  • Lemma 1: Gronwall Inequality albanez2024parameter
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2: Global well-posedness
  • proof
  • Theorem 3
  • proof