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

AbstractXMDS2 is a cross-platform, GPL-licensed, open source package for numerically integrating initial value problems that range froma single ordinary di erential equation up to systems of coupled stochastic partial di erential equations. The equations are describedin a high-level XML-based script, and the package generates low-level optionally parallelised C++ code for the ecient solutionof 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 amuch wider problem space while also producing faster code.Keywords: Initial value problems, di erential equations, numerical integration, stochastic partial di erential equationsProgram summary Manuscript Title: XMDS2: Fast, scalable simulation of coupledstochastic partial di erential equationsAuthors: Graham R. Dennis, Joseph J. Hope, Mattias T. JohnssonProgram Title: XMDS2Journal Reference:Catalogue identifier:Licensing provisions: GNU Public License 2Programming language: Python and C++.Computer: Any computer with a Unix-like system, a C++ compilerand PythonOperating system: Any Unix-like system; developed under Mac OS Xand GNU/LinuxRAM: Problem dependent (roughly 50 bytes per grid point)Number of processors used: Up to the minimum of the number ofpoints in each of the first two dimensionsKeywords: Initial value problems, di erential equations, stochasticpartial di erential equationsClassification: 4.3 Di erential Equations, 6.5 Software including Par-allel AlgorithmsExternal routines/libraries: The external libraries required areproblem-dependent. Uses FFTW3 Fourier transforms (used onlyfor FFT-based spectral methods), dSFMT random number generation(used only for stochastic problems), MPI message-passing interface(used only for distributed problems), HDF5, GNU Scientific Library(used only for Bessel-based spectral methods) and a BLAS implemen-tation (used only for non-FFT-based spectral methods).Nature of problem: General coupled initial-value stochastic partial dif-ferential equationsSolution method: Spectral method with method-of-lines integrationRunning time: Determined by the size of the problemWeb site: http://www.xmds.org

[1]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[2]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[3]  A. Carvalho,et al.  Feedback control of an interacting Bose-Einstein condensate using phase-contrast imaging , 2010, 1009.0587.

[4]  P. Lam,et al.  Effect of atomic noise on optical squeezing via polarization self-rotation in a thermal vapor cell , 2006 .

[5]  Jani-Petri Martikainen,et al.  Vortex dynamics in Bose-Einstein condensates , 2000 .

[6]  Makoto Matsumoto,et al.  SIMD-Oriented Fast Mersenne Twister: a 128-bit Pseudorandom Number Generator , 2008 .

[7]  Erwin Frey,et al.  Noise and correlations in a spatial population model with cyclic competition. , 2007, Physical review letters.

[8]  Ch. Tsitouras,et al.  Optimized explicit Runge-Kutta pair of orders 9(8) , 2001 .

[9]  Li Li,et al.  Pattern dynamics of a spatial predator–prey model with noise , 2012 .

[10]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[11]  Erwin Frey,et al.  Instability of spatial patterns and its ambiguous impact on species diversity. , 2008, Physical review letters.

[12]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[13]  A. H. Stroud,et al.  Numerical Quadrature and Solution of Ordinary Differential Equations: A Textbook for a Beginning Course in Numerical Analysis , 1974 .

[14]  T. Reichenbach,et al.  Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games , 2007, Nature.

[15]  Alan H. Karp,et al.  A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides , 1990, TOMS.

[16]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[17]  P. Lam,et al.  Characterization of electromagnetically-induced-transparency-based continuous-variable quantum memories , 2008 .

[18]  Peter D. Drummond,et al.  xmds: eXtensible multi-dimensional simulator , 2001 .

[19]  John L. Bohn,et al.  Bogoliubov modes of a dipolar condensate in a cylindrical trap (13 pages) , 2006 .

[20]  A. Truscott,et al.  Paired-atom laser beams created via four-wave mixing , 2008, 0806.0212.

[21]  R. J. Ballagh,et al.  Three-dimensional vortex dynamics in Bose-Einstein condensates , 2000 .

[22]  A. Truscott,et al.  Observation of transverse interference fringes on an atom laser beam. , 2007, Optics express.

[23]  Peter D. Drummond,et al.  Robust Algorithms for Solving Stochastic Partial Differential Equations , 1997 .