Energy-Stable, L2-Stable, and weakly H1-Stable BDF2 viscosity splitting method for the penalized incompressible navier-Stokes equations

초록

We develop an energy-stable, second-order accurate, and stable viscosity splitting method for the penalized incompressible Navier-Stokes equations. A time-dependent auxiliary variable and a time-dependent indicator function are introduced to design the numerical method. At each time step, it is sufficient to solve a Poisson equation to update the pressure and several linear elliptic equations with variable coefficients to update the velocity components. The complete decoupling of all computations enables an efficient numerical implementation. We analytically prove that the time-discretized energy satisfies the unconditional dissipation law. Moreover, we estimate that the L2-norm of velocity at each time step is bounded by the sum of the L2-norm of initial velocity and a constant, the H1-norm of velocity is weakly stable, i.e., it is bounded by a constant which is related to the time step. The accuracy and stability tests show that the proposed method has the expected second-order accuracy and energy dissipation property under different time steps. The numerical simulations of flow passing through a circular cylinder and a 3D airplane confirm the reliable performance of the proposed algorithm.

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