The PDE framework Peano applied to fluid dynamics: an efficient implementation of a parallel multiscale fluid dynamics solver on octree-like adaptive Cartesian grids

Abstract

This paper presents the general purpose framework Peano for the solution of partial differential equations (PDE) on adaptive Cartesian grids. The strict structuredness and inherent multilevel property of these grids allows for very low memory requirements, efficient (in terms of hardware performance) implementations of parallel multigrid solvers on dynamically adaptive grids, and arbitrary spatial dimensions. This combination of advantages distinguishes Peano from other PDE frameworks. We describe shortly the underlying octree-like grid type and its most important properties. The main part of the paper shows the framework concept of Peano and the implementation of a Navier-Stokes solver as one of the main currently implemented application examples. Various results ranging from hardware and numerical performance to concrete application scenarios close the contribution.

Navier–Stokes and Lattice–Boltzmann on octree‐like grids in the Peano framework

“…Thus, the Peano curve coming along with a tri-partitioning in grid refinement seems to be the only possible traversal for Cartesian grids allowing for the usage of a stack-based data storage. ‡ The L2-cache hitrate achieved with these methods has been above 98% for all applications tested in the Peano framework so far [14,15,20,21].…”

“…Peano provides defined events or, in other words, plug-in points, in the grid traversal algorithm where a user can implement own functionality [21,22]. For example, a Jacobi solver for systems of linear equations with vertex-centered degrees of freedom triggers the evaluation of the cellpart of the residual at the event enterElement and the updating of the solution at the event touchVertexLastTime.…”

“…The implementation of the NS equations in Peano is described in more detail in [21,25]. We only give a short overview of the main features of the NS solver.…”

Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework

“…The Lattice Boltzmann model and the respective slip and transition flow extensions are implemented within the Lattice Boltzmann application [11,32] of the Peano framework [33]. This framework provides adaptive Cartesian grid structures that are traversed along the iterates of the space-filling Peano curve.…”

A Scala prototype to generate multigrid solver implementations for different problems and target multi-core platforms

“…In practice multigrid algorithms are implemented in various frameworks, e. g. in WAL-BERLA for finite difference discretizations on fully structured grids also supporting GPU clusters [23], Boomer AMG [13] for unstructured grids and general matrices, Peano [8] that is based on space-filling curves, or DUNE [3] that is a general software framework for solving PDEs. Further examples of geometric multigrid solvers are found in [1] for an unstructured Finite Element (FE) elasticity and plasticity problem and for a variable-coefficient Poisson problem with a proposed matrix-free distributed octree geometric multigrid algorithm in [34] or an algebraic multigrid solver on GPUs [20].…”