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.

XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations

XMDS2 is a cross-platform, GPL-licensed, open source package for numerically integrating initial value problems that range from a single ordinary differential equation up to systems of coupled stochastic partial differential equations. The equations are described in a high-level XML-based script, and the package generates low-level optionally parallelised C++ code for the efficient solution of those equations. It combines the advantages of high-level simulations, namely fast and low-error development, with the speed, portability and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS package, and features support for a much wider problem space while also producing faster code.Comment: 9 pages, 5 figure

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Adaptive absorbing boundary conditions for Schrodinger-type equations: application to nonlinear and multi-dimensional problems

We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schr\"{o}dinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu and H. Han, {\it Phys. Rev. E}, 74(2006), pp. 037704] to problems in multidimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schr\"{o}dinger equations in one- and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.Comment: 18 pages; 12 figures. A short movie for the 2D NLS equation with absorbing boundary conditions can be downloaded at http://home.ustc.edu.cn/~xuzl/movie.avi. To appear in Journal of Computational Physic

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes.Comment: 29 page

Un método Wavelet-Galerkin para ecuaciones diferenciales parciales parabólicas

In this paper an Adaptive Wavelet-Galerkin method for the solution ofparabolic partial differential equations modeling physical problems withdifferent spatial and temporal scales is developed. A semi-implicit timedifference scheme is applied andB-spline multiresolution structure on theinterval is used. As in many cases these solutions are known to presentlocalized sharp gradients, local error estimators are designed and an ef-ficient adaptive strategy to choose the appropriate scale for each time isdeveloped. Finally, experiments were performed to illustrate the applica-bility and efficiency of the proposed method.En este trabajo se desarrolla un método Wavelet-Galerkin Adaptativopara la resolución de ecuaciones diferenciales parabólicas que modelanproblemas físicos, con diferentes escalas en el espacio y en el tiempo. Seutiliza un esquema semi-implícito en diferencias temporales y la estructuramultirresolución de las B-splines sobre intervalo.Como es sabido que enmuchos casos las soluciones presentan gradientes localmente altos, se handiseñado estimadores locales de error y una estrategia adaptativa eficientepara elegir la escala apropiada en cada tiempo. Finalmente, se realizaronexperimentos que ilustran la aplicabilidad y la eficiencia del método pro-puestoFil: Vampa, Victoria Cristina. Universidad Nacional de La Plata. Facultad de Ingeniería; ArgentinaFil: Martín, María Teresa. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ingeniería; Argentin