An element-based spectrally-optimized approximate inverse preconditioner for the Euler equations

Non-hydrostatic Unified Model of the Atmosphere (NUMA)The first NUMA papers appeared in 2008. From 2008 through 2010, all the NUMA papers appearing involved the 2D (x-z slice) Euler equations. All the theory and numerical implementations were first developed in 2D.We introduce a method for constructing an element-by-element sparse approximate inverse (SAI) preconditioner designed to be effective in a massively-parallel spectral element modeling environment involving non- symmetric systems. This new preconditioning approach is based on a spectral optimization of a low-resolution pre- conditioned system matrix (PSM). We show that the local preconditioning matrices obtained via this element-based, spectrum-optimized (EBSO) approach may be applied to arbitrarily high-resolution versions of the same system matrix without appreciable loss of preconditioner performance. We demonstrate the performance of the EBSO precondition- ing approach using 2-D spectral element method (SEM) formulations for a simple linear conservation law and for the fully-compressible 2-D Euler equations with various boundary conditions. For the latter model running at suffi- ciently large Courant number, the EBSO preconditioner significantly reduces both iteration count and wall-clock time regardless of whether a generalized minimum residual (GMRES) or a stabilized biconjugate gradient (BICGSTAB) iterative scheme is employed. To assess the value added by this new preconditioning approach, we compare its perfor- mance against two other equally-parallel SAI preconditioning methods: low-order Chebyshev generalized least-squares polynomials and an element-based variant of the well-known Frobenius norm optimization preconditioner which we also develop herein. The EBSO preconditioner significantly out-performs both the Chebyshev polynomials and the element-based Frobenius-norm-optimized (EBFO) preconditioner regardless of whether the GMRES or BICGSTAB iterative scheme is employed. Moreover, when the EBSO preconditioner is combined with the Chebyshev polynomial method dramatic reductions in iterations per time-step can be achieved while still achieving a significant reduction in wall-clock time

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

An adaptive preconditioner for steady incompressible flows

This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier--Stokes flows. The preconditioner maps the identity (no preconditioner) to the Stokes preconditioner (preconditioning by Laplacian) through a continuous parameter and is built on a first order Euler time-discretization scheme. The preconditioner is tested onto two fluid configurations: three-dimensional doubly diffusive convection and a reduced model of shear flows. In the former case, Stokes preconditioning works but a mixed preconditioner is preferred. In the latter case, the system of equation is split and solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications, this preconditioner is expected to help a wide range of studies

Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by B\"{o}rm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.Comment: 22 pages, 6 Figures, 2 Table

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential / capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes the resulting systems of nonlinear algebraic equations are solved with Newton's method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust and efficient. In particular no post-processing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1000 processors

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

In this work we exploit agglomeration based $h$-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature $h$-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element $L^2$ projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.Comment: 78 pages, 7 figure