We perform a numerical study of a two-component reaction–diffusion model. By using numerical continuation methods, combined with state-of-the-art sparse linear and eigenvalue solvers, we systematically compute steady state solutions and analyze their stability and relations in both two and three space dimensions. The approach gives a more reliable and complete picture than previous efforts based on time integration schemes and is also typically much more efficient in terms of computing time. We are therefore able to produce a rich bifurcation diagram showing a variety of solution patterns and transitions.

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We developed a finite volume package FVM and a solver HYMLS, both based on elements of the Trilinos EPETRA-package (see http://trilinos.sandia.gov/). HYMLS is a linear system solver for steady state incompressible Navier-Stokes equations coupled to transport equations in 2 and 3D [1, 2, 3]. We constructed recently a multilevel variant of it, which makes it possible to solve 3D problems of over 10 million unknowns quickly on a parallel computer. The behavior of the method is very much like that of multigrid methods. In fact one could see it as the father of the multigrid method. The solver is very robust. For the problem described in [4], it allowed a quick increase in the Reynolds number to get into the interesting region around Re=2000. Here we will show the performance of the method on the Rayleigh-Bénard convection in a cube, with six no-slip walls [5]. Also here we employ HYMLS to solve the linear systems resulting from a Cayley transform of the generalized eigenvalue problem.@en

We discuss the parallel implementation of a hybrid direct/iterative solver for a special class of saddle point matrices arising from the discretization of the steady Navier-Stokes equations on an Arakawa C-grid, the F-matrices. The two-level method described here has the following properties: (i) it is very robust, even at comparatively high Reynolds Numbers, (ii) a single parameter controls fill and convergence, making the method straightforward to use, (iii) the convergence rate is independent of the number of unknowns, (iv) it can be implemented on distributed memory machines in a natural way, (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively. The implementation focusses on generality, modularity, code reuse and recursiveness. The solver is implemented using building blocks of the Trilinos libraries. We show its performance on a parallel computer for the Navier-Stokes equations.@en

We present a new hybrid direct/iterative approach to the solution of a special class of saddle point matrices arising from the discretization of the steady incompressible Navier-Stokes equations on an Arakawa C-grid. The two-level method introduced here has the following properties: (i) it is very robust, even close to the point where the solution becomes unstable; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively; (vi) the iteration takes place in the divergence-free space, so the method qualifies as a "constraint preconditioner"; (vii) the approach can also be applied to Poisson problems. This work is also relevant for problems in which similar saddle point matrices occur, for instance, when simulating electrical networks, where one has to satisfy Kirchhoff's conservation law for currents.@en

In present-day forward time stepping ocean-climate models, capturing both the wind-driven and thermohaline components, a substantial amount of CPU time is needed in a so-called spin-up simulation to determine an equilibrium solution. In this paper, we present methodology based on Jacobian-Free Newton-Krylov methods to reduce the computational time for such a spin-up problem. We apply the method to an idealized configuration of a state-of-the-art ocean model, the Modular Ocean Model version 4 (MOM4). It is shown that a typical speed-up of a factor 10-25 with respect to the original MOM4 code can be achieved and that this speed-up increases with increasing horizontal resolution.@en

To understand the physics and dynamics of the ocean circulation, techniques of numerical bifurcation theory such as continuation methods have proved to be useful. Up to now these techniques have been applied to models with relatively few (O (105)) degrees of freedom such as multi-layer quasi-geostrophic and shallow-water models and relatively low-resolution (e.g., 4° horizontal resolution) primitive equation models. In this paper, we present a new approach in which continuation methods are combined with parallel numerical linear system solvers. With this implementation, we show that it is possible to compute steady states versus parameters (and perform fully implicit time integration) of primitive equation ocean models with up to a few million degrees of freedom.@en