Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations Original Research Article

This paper examines the long-term dissipativity and unconditional non-linear stability of time integration algorithms for the incompressible MHD equations. The results apply as a particular case to the uncoupled incompressible Navier-Stokes equations. The analysis is performed in the setting defined by an abstract evolution equation possessing the same long-term dissipative structure as the above physical systems. A general class of time-stepping schemes is analyzed in this context, showing the non-linear unconditional stability and long-term dissipativity of particular members of this family of algorithms. In particular, algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified. A particular linear algorithm is shown to exhibit optimal long-term dissipative properties and to possess an algorithmic global attractor. These abstract results apply rigorously to Galerkin mixed finite element discretizations of the problem. To this end, it is shown that such Galerkin-type projection inherits the dissipative properties of the continuum problem. Computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations.