Block Fusion on Dynamically Adaptive Spacetree Grids for Shallow Water Waves

Weinzierl, Tobias; Bäder, Michael; Unterweger, Kristof; Wittmann, R.

doi:10.1142/s0129626414410060

Cited by 13 publications

(31 citation statements)

References 10 publications

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“…For future work tackling the smoother challenge, we refer in particular to patch-based approaches [Ghysels et al 2013;Ghysels and Vanroose 2014]. Proof-of-concept studies from other application areas exist that use the same software infrastructure [Weinzierl et al 2014] to embed small regular Cartesian grids into each spacetree cell. These small grids, patches, allow for improved robustness due to stronger smoothers resulting from Chebyshev iterations, higher-order smoothing schemes on embedded regular grids or a multilevel Krylov solver based on recursive coarse grid deflation [Sheikh et al 2013;Erlangga and Nabben 2008].…”

Section: Discussionmentioning

confidence: 99%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based upon the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free.Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realise full approximation storage (FAS) within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realise a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as enabler for full multigrid (FMG) cycling-the grid unfolds throughout the computation-allows us to reduce the cost per unknown per order of accuracy.The present paper primary contributes towards software realisation and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR and vectorisation-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

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Section: Discussionmentioning

confidence: 99%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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“…Patch-based strategies [25,26,56], where patches of regular grids are embedded into cells, have been applied successfully for spacetrees and facilitate loop parallelism. Such approaches even can be generalized in a multiscale way, where whole regions are tessellated by a cascade of regular grids [27,28].…”

Section: Remarks On the Shared Memory Parallelizationmentioning

confidence: 99%

“…Furthermore, we note that the mapping of multigrid element activities onto tasks yields a high theoretical concurrency but also yields high task management overhead. To reduce this overhead and, hence, to increase the arithmetic intensity, our tasks have to be merged into bigger task assemblies [45,56]. The exact choice of the size of such mergers is a non-trivial task [14].…”

Section: Remarks On the Shared Memory Parallelizationmentioning

confidence: 99%

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Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

Weinzierl

2018

ACM Trans. Math. Softw.

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We present a family of spacetree-based multigrid realizations using the tree's multiscale nature to derive coarse grids. They align with matrix-free geometric multigrid solvers as they never assemble the system matrices which is cumbersome for dynamically adaptive grids and full multigrid. The most sophisticated realizations use BoxMG to construct operator-dependent prolongation and restriction in combination with Galerkin/Petrov-Galerkin coarse-grid operators. This yields robust solvers for nontrivial elliptic problems. We embed the algebraic, problem-and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in-situ throughout the grid traversals. While such an approach is not literally matrix-free-the grid carries the matrix-we propose to switch to a hierarchical representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be stored and exchanged with small memory footprint. Our realizations support arbitrary dynamically adaptive grids while they vertically integrate the multilevel operations through spacetree linearization. This yields good memory access characteristics, while standard colouring of mesh entities with domain decomposition allows us to use parallel manycore clusters. All realization ingredients are detailed such that they can be used by other codes.

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“…Very scalable and powerful schemes can be used, see e.g. [15]. But, the presence of a complex moving interface (composed of rarefaction, shocks and/or contacts) implies to re-mesh at each time step, which is obviously a costly process.…”

Section: Block Based Adaptive Mesh Refinement Procedures (Bb-amr)mentioning

confidence: 99%

Adaptive Mesh Refinement Method Applied to Shallow Water Model: A Mass Conservative Projection

Pons¹,

Golay²,

Marcer³

2017

Topical Problems of Fluid Mechanics 2017

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In the tsunami waves context, efficient numerical methods are necessary to simulate multi scales events. One way to reduce the computational cost is to use an adaptive mesh refinement method on unstructured meshes. This approach is used in this paper with a finite volume scheme solving the multi-dimensional Saint-Venant system. The adaptive mesh refinement method follows a block-based decomposition (called BB-AMR), which allows quick meshing and easy parallelization. One step of the AMR method is the so-called projection step where the new mesh values have to be defined from the old ones. For vertically integrated model the bathymetry variation during the projection step plays a crucial role on the mass conservation. To avoid large deficit or excess of mass a special attention is given to the projection step. Finally a 3D test case simulation is compared to experimental results to illustrated the quasi conservation of the mass with an appropriated projection method.

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Block Fusion on Dynamically Adaptive Spacetree Grids for Shallow Water Waves

Cited by 13 publications

References 10 publications

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

Adaptive Mesh Refinement Method Applied to Shallow Water Model: A Mass Conservative Projection

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