Multigrid and preconditioning strategies for implicit PDE solvers for degenerate parabolic equations
The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution.
The complex dynamics of an autonomous prey-predator system with impulsive state feedback control is studied. The sufficient conditions for the existence and stability of semi-trivial are obtained by using the Poincare map. Numerical simulations show that the complex dynamics of system we considered, including the doubling-bifurcations, chaos, etc