Time-parallel iterative solvers for parabolic evolution equations

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties

Space-Time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces

Semi-Lagrangian methods for parabolic problems in divergence form

Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection--diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches