Lagrange multiplier methods with energy dissipation for the incompressible Navier-Stokes equations

 This talk is concerned with the numerical solution of the Navier-Stokes equations, a fundamental system in computational fluid dynamics modeling the motion of incompressible viscous flow. One of the key questions when designing numerical methods for such equations is how to treat the nonlinear convection term. Fully implicit or semi-implicit schemes are often computationally costly, while implicit-explicit schemes require small time step sizes for numerical stability. Based on the Lagrange multiplier approach, we propose an efficient numerical method that allows substantially larger time steps while treating the nonlinear convection term explicitly. The idea is to incorporate the energy dissipation law into the system via a dynamic equation involving the kinetic energy, the Lagrange multiplier, and a regularization parameter. The reformulated problem is discretized in time based on the backward differentiation formulas, resulting in the so-called dynamically regularized Lagrange multiplier (DRLM) methods. DRLM schemes are unconditionally energy stable and require only the solutions of two linear Stokes systems (with constant coefficients) and a scalar quadratic equation at each time step. We establish optimal error estimates for the velocity and pressure (where the error bounds decay with respect to the regularization parameter) and present numerical results to illustrate the accuracy and robustness of the first- and second-order DRLM schemes.