662 research outputs found
Afivo: a framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods
Afivo is a framework for simulations with adaptive mesh refinement (AMR) on
quadtree (2D) and octree (3D) grids. The framework comes with a geometric
multigrid solver, shared-memory (OpenMP) parallelism and it supports output in
Silo and VTK file formats. Afivo can be used to efficiently simulate AMR
problems with up to about unknowns on desktops, workstations or single
compute nodes. For larger problems, existing distributed-memory frameworks are
better suited. The framework has no built-in functionality for specific physics
applications, so users have to implement their own numerical methods. The
included multigrid solver can be used to efficiently solve elliptic partial
differential equations such as Poisson's equation. Afivo's design was kept
simple, which in combination with the shared-memory parallelism facilitates
modification and experimentation with AMR algorithms. The framework was already
used to perform 3D simulations of streamer discharges, which required tens of
millions of cells
Unification of some advection schemes in two dimensions
The relationship between two approaches towards construction of genuinely two-dimensional upwind advection schemes is established. One of these approaches is of the control volume type applicable on structured cartesian meshes. It resulted in the compact high resolution schemes capable of maintaining second order accuracy in both homogeneous and inhomogeneous cases. Another one is the fluctuation splitting approach, which is well suited for triangular (and possibly) unstructured meshes. Understanding the relationship between these two approaches allows us to formulate here a new fluctuation splitting high resolution (i.e. possible use of artificial compression, while maintaining positivity property) scheme. This scheme is shown to be linearity preserving in inhomogeneous as well as homogeneous cases
Small Collaboration: Modeling Phenomena from Nature by Hyperbolic Partial Differential Equations (hybrid meeting)
Nonlinear hyperbolic partial differential equations constitute a plethora of models from physics, biology, engineering, etc. In this workshop we cover the range from modeling, mathematical questions of well-posedness, numerical discretization and numerical simulations to compare with the phenomenon from nature that was modeled in the first place. Both kinetic and fluid models were discussed
High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs
Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching
Multi-Dimensional Astrophysical Structural and Dynamical Analysis I. Development of a Nonlinear Finite Element Approach
A new field of numerical astrophysics is introduced which addresses the
solution of large, multidimensional structural or slowly-evolving problems
(rotating stars, interacting binaries, thick advective accretion disks, four
dimensional spacetimes, etc.). The technique employed is the Finite Element
Method (FEM), commonly used to solve engineering structural problems. The
approach developed herein has the following key features:
1. The computational mesh can extend into the time dimension, as well as
space, perhaps only a few cells, or throughout spacetime.
2. Virtually all equations describing the astrophysics of continuous media,
including the field equations, can be written in a compact form similar to that
routinely solved by most engineering finite element codes.
3. The transformations that occur naturally in the four-dimensional FEM
possess both coordinate and boost features, such that
(a) although the computational mesh may have a complex, non-analytic,
curvilinear structure, the physical equations still can be written in a simple
coordinate system independent of the mesh geometry.
(b) if the mesh has a complex flow velocity with respect to coordinate space,
the transformations will form the proper arbitrary Lagrangian- Eulerian
advective derivatives automatically.
4. The complex difference equations on the arbitrary curvilinear grid are
generated automatically from encoded differential equations.
This first paper concentrates on developing a robust and widely-applicable
set of techniques using the nonlinear FEM and presents some examples.Comment: 28 pages, 9 figures; added integral boundary conditions, allowing
very rapidly-rotating stars; accepted for publication in Ap.
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