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<p>This paper presents a first-order approximation for solving the unsteady Stokes problem using the pressure-correction Goda scheme in the Reduced Order Model (ROM) context. The study begins by formulating a semi-discrete in time approximation of the Stokes problem, followed by the introduction of the Goda time-splitting scheme, employing the first-order Euler approximation for time discretization. A brief overview of the Finite Element (FE) space approximation is also provided. The Proper Orthogonal Decomposition (POD) method is then used to construct a Galerkin projection-based ROM for the Goda equations. An interesting point of the present contribution lies in the choice of inner products used to construct reduced bases for predicted velocities, divergence-free velocities, and pressure. These choices lead to explicit calculations for deriving velocity and pressure solutions within the reduced model. Stability analysis and error estimates for the proposed ROM are derived. Finally, numerical examples are presented to validate the theoretical results, demonstrating the effectiveness and accuracy of the proposed approach also for the simulation of the parametrized incompressible Navier–Stokes equations.</p>

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A POD-Based Reduced Order Method Applied to Goda Time-Splitting Scheme

  • 期刊论文

  • 发表日期：

    2025年5月19日

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  • M. Azaïez,
  • T. Chacón Rebollo,
  • C. Núñez Fernández,
  • S. Rubino

摘要

This paper presents a first-order approximation for solving the unsteady Stokes problem using the pressure-correction Goda scheme in the Reduced Order Model (ROM) context. The study begins by formulating a semi-discrete in time approximation of the Stokes problem, followed by the introduction of the Goda time-splitting scheme, employing the first-order Euler approximation for time discretization. A brief overview of the Finite Element (FE) space approximation is also provided. The Proper Orthogonal Decomposition (POD) method is then used to construct a Galerkin projection-based ROM for the Goda equations. An interesting point of the present contribution lies in the choice of inner products used to construct reduced bases for predicted velocities, divergence-free velocities, and pressure. These choices lead to explicit calculations for deriving velocity and pressure solutions within the reduced model. Stability analysis and error estimates for the proposed ROM are derived. Finally, numerical examples are presented to validate the theoretical results, demonstrating the effectiveness and accuracy of the proposed approach also for the simulation of the parametrized incompressible Navier–Stokes equations.