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Scalability of Elliptic Solvers in Numerical Weather and Climate Prediction

Wednesday 24th October 2012 - 14:15 to 14:40
Numerical weather- and climate- prediction requires the solution of elliptic partial differential equations with a large number of unknowns on a spherical grid. In particular, if implicit time stepping is used in the dynamical core of the forecast model, an elliptic PDE has to be solved at each time step. This often amounts to a significant proportion of the model runtime. The goal of the Next Generation Weather and Climate Prediction (NGWCP) project is the development of a new dynamical core for the UK Met Office Unified Model with a significantly increased global model resolution, resulting in more than $10^{10}$ degrees of freedom for each atmospheric variable. To run the model operationally, the solver has to scale to hundreds of thousands of processor cores on modern computer architectures.

To investigate the scalability of the implicit time stepping algorithm we have tested and optimised existing solvers in the Distributed and Unified Numerics Environment (DUNE) and the Hypre library. In addition we also implemented a matrix-free parallel geometric multigrid code with a vertical line smoother. We demonstrate the scalability of the solvers on up to 65536 cores of the Hector supercomputer for a system with $10^{10}$ degrees of freedom for the elliptic PDE arising from semi-implicit semi-Lagrangian time stepping.

To identify the most promising solver we investigated the robustness of simple and widely used preconditioners, such as vertical line relaxation, and more advanced multigrid methods. We compared algebraic- and matrix-free geometric multigrid algorithms to quantify the matrix- and coarse-grid- setup costs and studied the performance of various solvers on different computer architectures.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons