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Non-intrusive hierarchical coupling strategies for multi-scale simulations in gravitational dynamics. (English) Zbl 1454.85005

Summary: Hierarchical code coupling strategies make it possible to combine the results of individual numerical solvers into a self-consistent symplectic solution. We explore the possibility of allowing such a coupling strategy to be non-intrusive. In that case, the underlying numerical implementation is not affected by the coupling itself, but its functionality is carried over in the interface. This method is efficient for solving the equations of motion for a self-gravitating system over a wide range of scales. We adopt a dedicated integrator for solving each particular part of the problem and combine the results to a self-consistent solution. In particular, we explore the possibilities of combining the evolution of one or more microscopic systems that are embedded in a macroscopic system. The here presented generalizations of Bridge include higher-order coupling strategies (from the classic 2nd order up to 10th-order), but we also demonstrate how multiple bridges can be nested and how additional processes can be introduced at the bridge time-step to enrich the physics, for example by incorporating dissipative processesor. Such augmentation allows for including additional processes in a classic Newtonian \(N\)-body integrator without alterations to the underlying code. These additional processes include for example the Yarkovsky effect, dynamical friction or relativistic dynamics. Some of these processes operate on all particles whereas others apply only to a subset.
The presented method is non-intrusive in the sense that the underlying methods remain operational without changes to the code (apart from adding the get- and set-functions to enable the bridge operator). As a result, the fundamental integrators continue to operate with their internal time step and preserve their local optimizations and parallelism. Multiple bridges can be nested and coupled hierarchically, allowing for the construction of a complex environment of multiple nested augmented bridges. While the coupling topology may become rather complicated, we introduce the hierarchical coupling language (HCL), a meta language in which complex bridge topologies can be described. The meta language is meant for stimulating the discussion on even more complex hierarchies in which the bridge operators are introduced as patterns. We present example applications for several of these cases and discuss the conditions under which these integrators can be applied. Typical applications range over 10 orders of magnitude in temporal and spatial scales when we apply the method to simulating planetary systems (AU spatial and year-temporal scale) in a star cluster that orbits in the Galaxy (100 kpc-spatial and 10 Gyr-temporal scale).

MSC:

85A15 Galactic and stellar structure
85-10 Mathematical modeling or simulation for problems pertaining to astronomy and astrophysics
70F15 Celestial mechanics
85-08 Computational methods for problems pertaining to astronomy and astrophysics
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
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