Original-energy-dissipation-preserving methods for the incompressible Navier-Stokes equations

TL;DR

This paper develops new energy-dissipation-preserving numerical schemes for the incompressible Navier-Stokes equations, ensuring energy law conservation at each step and demonstrating their efficiency and accuracy through rigorous proofs and experiments.

Contribution

It introduces novel structure-preserving schemes that exactly conserve the energy dissipation law for Navier-Stokes equations, using Crank-Nicolson and BDF methods with proven stability and efficiency.

Findings

Schemes exactly preserve energy dissipation law.

Methods require solving only linear systems.

Numerical experiments confirm accuracy and efficiency.

Abstract

This paper introduces a robust reformulation of the incompressible Navier-Stokes equations, establishing a foundational framework for designing efficient, structure-preserving algorithms that strictly conserve the original energy dissipation law. By leveraging Crank-Nicolson schemes and backward differentiation formulas, we develop four first- and second-order time-discrete schemes. These schemes exactly preserve the original energy dissipation law at each time step, requiring only the solutions of three linear Stokes systems and one $2\times 2$ system of linear equations. Furthermore, the finite difference approximation on a staggered grid is employed for these time-discrete systems to derive fully discrete structure-preserving schemes. We rigorously prove that all proposed fully discrete methods both maintain the original energy dissipation law and admit unique solutions. Moreover, we present their efficient implementation. Extensive numerical experiments are carried out to verify the accuracy, efficacy, and advantageous performance of our newly developed methods.