Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations

Abstract

This paper examines the long-term behavior, dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible Navier-Stokes equations, including both direct schemes and fractional step/projection methods. The algorithms are termed nonlinear unconditionally stable if, in the absence of a forcing term but for arbitrary initial conditions, the computed kinetic energy decreases for arbitrary step sizes (i.e., L 2-stability). Such an a priori stability estimate is a characteristic feature of the Navier-Stokes system. Similarly, the algorithms are said to exhibit asymptotic δ t-independent long-term dissipative behavior if, as the continuum flow generated by the Navier-Stokes equations, the algorithmic flow also exhibits an absorbing set (and a maximal attractor). Both direct and fractional step methods are described which inherit these fundamental properties of the Navier-Stokes system for any time step size. Moreover, a subclass of algorithms which retain these strong notions of nonlinear stability and long-term dissipative behavior is identified which, in addition, has the remarkable property of being linear within the time step. The key implication is that no iterations are required to compute the solution within a time step. A specific member of this class is shown to possess the same absorbing set as the continuum Navier-Stokes system. A second order accurate, unconditionally stable method is also identified which retains the property of linearity within a time step. Numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations.