11,642 research outputs found
New versions of iterative splitting methods for the momentum equation
[EN] In this paper we propose some modifications in the schemes for the iterative splitting techniques defined in Geiser (2009) for partial differential equations and introduce the parallel version of these modified algorithms. Theoretical results related to the order of the iterative splitting for these schemes are obtained. In the numerical experiments we compare the obtained results by applying iterative methods to approximate the solutions of the nonlinear systems obtained from the discretization of the splitting techniques to the mixed convection-diffusion Burgers' equation and a momentum equation that models a viscous flow. The differential equations in each splitting interval are solved by the back-Euler-Newton algorithm using sparse matrices. (C) 2016 Elsevier B.V. All rights reserved.This work has been supported by Ministerio de EconomĂa y Competitividad de España MTM2014-52016-02-2-PGeiser, J.; Hueso Pagoaga, JL.; MartĂnez Molada, E. (2017). New versions of iterative splitting methods for the momentum equation. Journal of Computational and Applied Mathematics. 309:359-370. https://doi.org/10.1016/j.cam.2016.06.002S35937030
Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations
[EN] This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection-diffusion-reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers' and Maxwell-Stefan diffusion equations.This research was funded by German Academic Exchange Service grant number 91588469.
We acknowledge support by the DFG Open Access Publication Funds of the Ruhr-Universität of
Bochum, Germany and by Ministerio de EconomĂa y Competitividad, Spain, under grant PGC2018-095896-B-C21-C22.Geiser, J.; Hueso, JL.; MartĂnez Molada, E. (2020). Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations. Mathematics. 8(3):1-22. https://doi.org/10.3390/math8030302S12283Auzinger, W., & Herfort, W. (2014). Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opuscula Mathematica, 34(2), 243. doi:10.7494/opmath.2014.34.2.243Auzinger, W., Koch, O., & Quell, M. (2016). Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions. Numerical Algorithms, 75(1), 261-283. doi:10.1007/s11075-016-0206-8Descombes, S., & Massot, M. (2004). Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction. Numerische Mathematik, 97(4), 667-698. doi:10.1007/s00211-003-0496-3Descombes, S., Dumont, T., Louvet, V., & Massot, M. (2007). On the local and global errors of splitting approximations of reaction–diffusion equations with high spatial gradients. International Journal of Computer Mathematics, 84(6), 749-765. doi:10.1080/00207160701458716McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053Trotter, H. F. (1959). On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4), 545-545. doi:10.1090/s0002-9939-1959-0108732-6Strang, G. (1968). On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517. doi:10.1137/0705041Jahnke, T., & Lubich, C. (2000). Bit Numerical Mathematics, 40(4), 735-744. doi:10.1023/a:1022396519656Nevanlinna, O. (1989). Remarks on Picard-Lindelöf iteration. BIT, 29(2), 328-346. doi:10.1007/bf01952687Farago, I., & Geiser, J. (2007). Iterative operator-splitting methods for linear problems. International Journal of Computational Science and Engineering, 3(4), 255. doi:10.1504/ijcse.2007.018264DESCOMBES, S., DUARTE, M., DUMONT, T., LOUVET, V., & MASSOT, M. (2011). ADAPTIVE TIME SPLITTING METHOD FOR MULTI-SCALE EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATIONS. Confluentes Mathematici, 03(03), 413-443. doi:10.1142/s1793744211000412Geiser, J. (2008). Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations. Journal of Computational and Applied Mathematics, 217(1), 227-242. doi:10.1016/j.cam.2007.06.028Dimov, I., Farago, I., Havasi, A., & Zlatev, Z. (2008). Different splitting techniques with application to air pollution models. International Journal of Environment and Pollution, 32(2), 174. doi:10.1504/ijep.2008.017102Karlsen, K. H., Lie, K.-A., Natvig, J. ., Nordhaug, H. ., & Dahle, H. . (2001). Operator Splitting Methods for Systems of Convection–Diffusion Equations: Nonlinear Error Mechanisms and Correction Strategies. Journal of Computational Physics, 173(2), 636-663. doi:10.1006/jcph.2001.6901Geiser, J. (2010). Iterative operator-splitting methods for nonlinear differential equations and applications. Numerical Methods for Partial Differential Equations, 27(5), 1026-1054. doi:10.1002/num.20568Geiser, J., & Wu, Y. H. (2015). Iterative solvers for the Maxwell–Stefan diffusion equations: Methods and applications in plasma and particle transport. Cogent Mathematics, 2(1), 1092913. doi:10.1080/23311835.2015.1092913Geiser, J., Hueso, J. L., & MartĂnez, E. (2017). New versions of iterative splitting methods for the momentum equation. Journal of Computational and Applied Mathematics, 309, 359-370. doi:10.1016/j.cam.2016.06.002Boudin, L., Grec, B., & Salvarani, F. (2012). A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete & Continuous Dynamical Systems - B, 17(5), 1427-1440. doi:10.3934/dcdsb.2012.17.1427Duncan, J. B., & Toor, H. L. (1962). An experimental study of three component gas diffusion. AIChE Journal, 8(1), 38-41. doi:10.1002/aic.69008011
The stransverse mass, MT2, in special cases
This document describes some special cases in which the stransverse mass,
MT2, may be calculated by non-iterative algorithms. The most notable special
case is that in which the visible particles and the hypothesised invisible
particles are massless -- a situation relevant to its current usage in the
Large Hadron Collider as a discovery variable, and a situation for which no
analytic answer was previously known. We also derive an expression for MT2 in
another set of new (though arguably less interesting) special cases in which
the missing transverse momentum must point parallel or anti parallel to the
visible momentum sum. In addition, we find new derivations for already known
MT2 solutions in a manner that maintains manifest contralinear boost invariance
throughout, providing new insights into old results. Along the way, we stumble
across some unexpected results and make conjectures relating to geometric forms
of M_eff and H_T and their relationship to MT2.Comment: 11 pages, no figures. v2 corrects minor typos. v3 corrects an
incorrect statement in footnote 8 and inserts a missing term in eq (3.9). v4
and v5 correct minor typos spotted by reader
Implementation of the LANS-alpha turbulence model in a primitive equation ocean model
This paper presents the first numerical implementation and tests of the
Lagrangian-averaged Navier-Stokes-alpha (LANS-alpha) turbulence model in a
primitive equation ocean model. The ocean model in which we work is the Los
Alamos Parallel Ocean Program (POP); we refer to POP and our implementation of
LANS-alpha as POP-alpha. Two versions of POP-alpha are presented: the full
POP-alpha algorithm is derived from the LANS-alpha primitive equations, but
requires a nested iteration that makes it too slow for practical simulations; a
reduced POP-alpha algorithm is proposed, which lacks the nested iteration and
is two to three times faster than the full algorithm. The reduced algorithm
does not follow from a formal derivation of the LANS-alpha model equations.
Despite this, simulations of the reduced algorithm are nearly identical to the
full algorithm, as judged by globally averaged temperature and kinetic energy,
and snapshots of temperature and velocity fields. Both POP-alpha algorithms can
run stably with longer timesteps than standard POP.
Comparison of implementations of full and reduced POP-alpha algorithms are
made within an idealized test problem that captures some aspects of the
Antarctic Circumpolar Current, a problem in which baroclinic instability is
prominent. Both POP-alpha algorithms produce statistics that resemble
higher-resolution simulations of standard POP.
A linear stability analysis shows that both the full and reduced POP-alpha
algorithms benefit from the way the LANS-alpha equations take into account the
effects of the small scales on the large. Both algorithms (1) are stable; (2)
make the Rossby Radius effectively larger; and (3) slow down Rossby and gravity
waves.Comment: Submitted to J. Computational Physics March 21, 200
The Hanle Effect in 1D, 2D and 3D
This paper addresses the problem of scattering line polarization and the
Hanle effect in one-dimensional (1D), two-dimensional (2D) and
three-dimensional (3D) media for the case of a two-level model atom without
lower-level polarization and assuming complete frequency redistribution. The
theoretical framework chosen for its formulation is the QED theory of Landi
Degl'Innocenti (1983), which specifies the excitation state of the atoms in
terms of the irreducible tensor components of the atomic density matrix. The
self-consistent values of these density-matrix elements is to be determined by
solving jointly the kinetic and radiative transfer equations for the Stokes
parameters. We show how to achieve this by generalizing to Non-LTE polarization
transfer the Jacobi-based ALI method of Olson et al. (1986) and the iterative
schemes based on Gauss-Seidel iteration of Trujillo Bueno and Fabiani Bendicho
(1995). These methods essentially maintain the simplicity of the
Lambda-iteration method, but their convergence rate is extremely high. Finally,
some 1D and 2D model calculations are presented that illustrate the effect of
horizontal atmospheric inhomogeneities on magnetic and non-magnetic resonance
line polarization signals.Comment: 14 pages and 5 figure
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