3.1 Simulation: grid, data and communication management

For our simulations, the grid management has to allow adaptivity during runtime due to demands on the resolution to be increased by refining the grid in feature-rich areas (e.g. close to the wave front) and coarsening the grid in areas which do not have a large contribution to the final result.

The shallow water equations are frequently used models to research wave-propagations and a simulation of them only requires a two-dimensional domain discretization. Therefore, we focus on efficient simulations of two-dimensional scenarios with triangles as the basic primitives. Despite its two-dimensional nature, particular domain triangulations can be created to assemble a two-dimensional surface on a sphere (see Sec. 6.5). Using multiple layers in each cell, this leads to further possible applications such as weather simulations (see Sec. 6.6).

For the simulation of our considered hyperbolic equations, we particularly focus on the following requirements:

• Data access: To advance the simulation in time, data in other cells has to be made accessible, e.g. to flux limiters via edge- or node-based communication.

  We developed clear interfaces (Section 4.9) yielding efficient parallel communication schemes (Sections 5.2)

• Usability and parallelization: For usability reasons, a parallelization approach should lead to a grid, data and communication management which is almost transparent to the application developer.

  A multi-layer framework design was chosen to provide appropriate abstraction levels and extendibility (Section 5.3).

• Flexible simulation domain: Under consideration of the communication schemes developed in this work, we are able to generate domains which can be assembled with triangle primitives.

  Different kinds of domain triangulations such as those required for a simulation on a sphere can then be assembled (Section 5.4).

• Flexible cell traversals: For several scenarios, not all cells of the simulation domain require to be traversed. A well-known example is the optimization of the execution of iterative solvers. Additional smoother iterations on grid areas with a residual already undershooting a threshold can be avoided, yielding a local relaxation method [Rüd93]. Another optimization is given for the generation of a conforming grid without hanging nodes. Here, areas with an already conforming grid do not require further execution of traversals generating a conforming grid.

  These issues require sophisticated cell traversals and a software design offering the corresponding flexibility. We provide a possible solution by introducing clustering on dynamic adaptive grids and show results for the skipping of cluster traversals for creating a conforming grid, see e.g. Section 5.8.2