Métodos multimalla geométricos en mallas semi-estructuradas de Vorono

En este proyecto se presenta un metodo de discretización de ecuaciones en derivadas parciales en mallas triangulares semi-estructuradas usando volumenes finítos y como punto representativo el punto de Voronoi. La posterior discretización se resualve usando metodos multimalla semi-estructurados y se presentan un conjunto de nuevos suavizadores asi como un algoritmo de Galerkin de tipo RAP para cuando las condiciones no son homogeneas en toda la superficie. Finalmente se muestran un conjunto de ejemplo numéricos para demostrar los resultados obtenidos

Adaptive control in rollforward recovery for extreme scale multigrid

With the increasing number of compute components, failures in future exa-scale computer systems are expected to become more frequent. This motivates the study of novel resilience techniques. Here, we extend a recently proposed algorithm-based recovery method for multigrid iterations by introducing an adaptive control. After a fault, the healthy part of the system continues the iterative solution process, while the solution in the faulty domain is re-constructed by an asynchronous on-line recovery. The computations in both the faulty and healthy subdomains must be coordinated in a sensitive way, in particular, both under and over-solving must be avoided. Both of these waste computational resources and will therefore increase the overall time-to-solution. To control the local recovery and guarantee an optimal re-coupling, we introduce a stopping criterion based on a mathematical error estimator. It involves hierarchical weighted sums of residuals within the context of uniformly refined meshes and is well-suited in the context of parallel high-performance computing. The re-coupling process is steered by local contributions of the error estimator. We propose and compare two criteria which differ in their weights. Failure scenarios when solving up to $6.9\cdot10^{11}$ unknowns on more than 245\,766 parallel processes will be reported on a state-of-the-art peta-scale supercomputer demonstrating the robustness of the method

Non-invasive multigrid for semi-structured grids

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid

Efficient parallel 3D computation of the compressible Euler equations with an invariant-domain preserving second-order finite-element scheme

We discuss the efficient implementation of a high-performance second-order collocation-type finite-element scheme for solving the compressible Euler equations of gas dynamics on unstructured meshes. The solver is based on the convex limiting technique introduced by Guermond et al. (SIAM J. Sci. Comput. 40, A3211-A3239, 2018). As such it is invariant-domain preserving, i.e., the solver maintains important physical invariants and is guaranteed to be stable without the use of ad-hoc tuning parameters. This stability comes at the expense of a significantly more involved algorithmic structure that renders conventional high-performance discretizations challenging. We develop an algorithmic design that allows SIMD vectorization of the compute kernel, identify the main ingredients for a good node-level performance, and report excellent weak and strong scaling of a hybrid thread/MPI parallelization

Entwicklung und Anwendung von Hochleistungs-Software für Mantelkonvektionssimulationen

The Earth mantle convects on a global scale, coupling the stress field at every point to every other location at an instant. This way, any change in the buoyancy field has an immediate impact on the convection patterns worldwide. At the same time, mantle convection couples to processes at scales of a few kilometers or even a few hundred meters. Dynamic topography and the geoid are examples of such small-scale expressions of mantle convection. Also, the depth of phase transitions varies locally, with strong influences on the buoyancy, and thus the global stress field. In order to understand these processes dynamically it is essential to resolve the whole mantle at very high numerical resolutions. At the same time, geodynamicists are trying to answer new questions with their models, for example about the rheology of the mantle, which is most likely highly nonlinear. Also, due to the extremely long timescales we cannot observe past mantle states, which calls for simulations backwards in time. All these issues lead to an extreme demand in computing power. To cater to those needs, the physical models of the mantle have to be matched with efficient solvers and fast algorithms, such that we can efficiently exploit the enormous computing power of current and future high performance systems. Here, we first give an extensive overview over the physical models and introduce some numerical concepts to solve the equations. We present a new two-dimensional software as a testbed and elaborate on the implications of realistic mineralogic models for efficient mantle convection simulations. We find that phase transitions present a major challenge and suggest some procedures to incorporate them into mantle convection modeling. Then we give an introduction to the high-performance mantle convection prototype HHG, a multigrid-based software framework that scales to some of the fastest computers currently available. We adapt this framework to a spherical geometry and present first application examples to answer geodynamic questions. In particular, we show that a very thin and very weak asthenosphere is dynamically plausible and consistent with direct and indirect geological observations.Englische Übersetzung des Titels: Development and application of high performance software for mantle convection modelin