Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

Abstract

Adaptive multiscale methods are among the many effective techniques for the numerical solution of partial differential equations. Efficient grid management is an important task in these solvers. In this paper we focus on this problem for discontinuous Galerkin discretization methods in 2 and 3 spatial dimensions and present a data structure for handling adaptive grids of different cell types in a unified approach. Instead of tree-based techniques where connectivity is stored via pointers, we associate each cell that arises in the refinement hierarchy with a cell identifier and construct algorithms that establish hierarchical and spatial connectivity. By means of bitwise operations, the complexity of the connectivity algorithms can be bounded independent of the level. The grid is represented by a hash table which results in a low-memory data structure and ensures fast access to cell data. The spatial connectivity algorithm also supports the application of quadrature rules for face integrals that occur in discontinuous Galerkin discretizations. The concept allows us to implement discontinuous Galerkin methods largely independent of spatial dimension and cell type. We demonstrate this by outlining how typical algorithmic tasks that arise in these implementations can be performed with our data structure. In computational tests we compare our approach with that of a classical implementation using pointers.

New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators

“…A dynamic graded tree structure is used in this implementation to represent data in the computer memory. This kind of data structure has been used in other multiresolution applications [28] and other dedicated data structures have also been developed [3]. The adapted grid corresponds to a set of nested dyadic grids generated by refining recursively a given cell, depending on the local regularity of the solution.…”

“…Therefore, an important amount of work is still in progress concerning programming features such as data structures, optimized routines and parallelization strategies for the time integration technique as well as for the multiresolution environment, even though the global CPU time is largely dominated by the time resolution for these stiff problems. For instance, some dedicated and efficient implementations have been recently developed for multiresolution applications [3,4]. Finally, when dealing with more complex systems such as complex or more detailed chemistry or stroke modeling in the brain, the source term involves many species (typically 50) and many reactions (typically several hundreds) or complex mechanisms.…”

Adaptive Multiresolution Methods: Practical issues on Data Structures, Implementation and Parallelization

“…In case of triangulations, the supports cannot be described in such a compact form. Instead we have to apply the neighboring algorithms provided in [10] to compute the sets during runtime.…”

“…The refinement rule defines how to divide a given cell into its child cells, cf. [10]. Thus, each cell that may occur within the grid hierarchy is characterized by a cell identifier, which will also provide the hierarchical connectivity.…”

“…It has been recently extended with respect to non-Cartesian grid hierarchies, see [48]. In [10], a unified theory is established that shows under which conditions imposed on the grids and refinement rules a sparse data structure can be constructed, that efficiently manages adaptive multilevel grids consisting of triangular and quadrilateral cells (in 2D) and tetrahedral, cuboid and prismatic cells (in 3D) and also hybrid grids composed of different cell types. In view of an optimal memory management and a fast data access the well-known concept of hash maps is used.…”