1,599 research outputs found
On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity
The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used
in numerical relativity for the solution of both hyperbolic and parabolic
partial differential equations. We here extend the recent work on the stability
of this scheme for hyperbolic equations by investigating the properties when
the average between the predicted and corrected values is made with unequal
weights and when the scheme is applied to a parabolic equation. We also propose
a variant of the scheme in which the coefficients in the averages are swapped
between two corrections leading to systematically larger amplification factors
and to a smaller numerical dispersion.Comment: 7 pages, 3 figure
Evaluating the ENVI-met microscale model for suitability in analysis of targeted urban heat mitigation strategies
Microscale atmospheric models are increasingly being used to project the thermal benefits of urban heat mitigation strategies (e.g., tree planting programs or use of high-albedo materials). However, prior to investment in specific mitigation efforts by local governments, it is desirable to test and validate the computational models used to evaluate strategies. While some prior studies have conducted limited evaluations of the ENVI-met microscale climate model for specific case studies, there has been relatively little systematic testing of the model's sensitivity to variations in model input and control parameters. This study builds on the limited foundation of past validation efforts by addressing two questions: (1) is ENVI-met grid independent; and (2) can the model adequately represent the air temperature perturbations associated with heat mitigation strategies? To test grid independence, a âflatâ domain is tested with six vertical grid resolutions ranging from 0.75 to 2.0âŻm. To examine the second question, a control and two mitigation strategy simulations of idealized city blocks are tested. Results show a failure of grid independence in the âflatâ domain simulations. Given that the mitigation strategies result in temperature changes that are an order of magnitude larger than the errors introduced by grid dependence for the flat domain, a lack of grid independence itself does not necessarily invalidate the use of ENVI-met for heat mitigation research. However, due to limitations in grid structure of the ENVI-met model, it was not possible to test grid dependence for more complicated simulations involving domains with buildings. Furthermore, it remains unclear whether existing efforts at model validation provide any assurance that the model adequately captures vertical mixing and exchange of heat from the ground to rooftop level. Thus, there remain concerns regarding the usefulness of the model for evaluating heat mitigation strategies, particularly when applied at roof level (e.g. high albedo or vegetated roofs)
Reaction-Diffusion Process Driven by a Localized Source: First Passage Properties
We study a reaction-diffusion process that involves two species of atoms,
immobile and diffusing. We assume that initially only immobile atoms, uniformly
distributed throughout the entire space, are present. Diffusing atoms are
injected at the origin by a source which is turned on at time t=0. When a
diffusing atom collides with an immobile atom, the two atoms form an immobile
stable molecule. The region occupied by molecules is asymptotically spherical
with radius growing as t^{1/d} in d>=2 dimensions. We investigate the survival
probability that a diffusing atom has not become a part of a molecule during
the time interval t after its injection and the probability density of such a
particle. We show that asymptotically the survival probability (i) saturates in
one dimension, (ii) vanishes algebraically with time in two dimensions (with
exponent being a function of the dimensionless flux and determined as a zero of
a confluent hypergeometric function), and (iii) exhibits a stretched
exponential decay in three dimensions.Comment: 7 pages; version 2: section IV is re-written, references added, 8
pages (final version
Anomalous Scaling and Solitary Waves in Systems with Non-Linear Diffusion
We study a non-linear convective-diffusive equation, local in space and time,
which has its background in the dynamics of the thickness of a wetting film.
The presence of a non-linear diffusion predicts the existence of fronts as well
as shock fronts. Despite the absence of memory effects, solutions in the case
of pure non-linear diffusion exhibit an anomalous sub-diffusive scaling. Due to
a balance between non-linear diffusion and convection we, in particular, show
that solitary waves appear. For large times they merge into a single solitary
wave exhibiting a topological stability. Even though our results concern a
specific equation, numerical simulations supports the view that anomalous
diffusion and the solitary waves disclosed will be general features in such
non-linear convective-diffusive dynamics.Comment: Corrected typos, added 3 references and 2 figure
Solving two-phase freezing Stefan problems: Stability and monotonicity
[EN] The two-phase Stefan problems with phase formation and depletion are special
cases ofmoving boundary problemswith interest in science and industry. In this
work, we study a solidification problem, introducing a front-fixing transformation.
The resulting non-linear partial differential system involves singularities,
both at the beginning of the freezing process and when the depletion is complete,
that are treated with special attention in the numerical modelling. The
problem is decomposed in three stages, in which implicit and explicit finite
difference schemes are used. Numerical analysis reveals qualitative properties
of the numerical solution spatial monotonicity of both solid and liquid temperatures
and the evolution of the solidification front. Numerical experiments
illustrate the behaviour of the temperatures profiles with time, as well as the
dynamics of the solidification front.Ministerio de Ciencia, Innovacion y Universidades, Grant/Award Number: MTM2017-89664-P.Piqueras, MA.; Company Rossi, R.; JĂłdar SĂĄnchez, LA. (2020). Solving two-phase freezing Stefan problems: Stability and monotonicity. Mathematical Methods in the Applied Sciences. 43(14):7948-7960. https://doi.org/10.1002/mma.5787S794879604314Schmidt, A. (1996). Computation of Three Dimensional Dendrites with Finite Elements. Journal of Computational Physics, 125(2), 293-312. doi:10.1006/jcph.1996.0095Singh, S., & Bhargava, R. (2014). Simulation of Phase Transition During Cryosurgical Treatment of a Tumor Tissue Loaded With Nanoparticles Using Meshfree Approach. Journal of Heat Transfer, 136(12). doi:10.1115/1.4028730Company, R., Egorova, V. N., & JĂłdar, L. (2014). Solving American Option Pricing Models by the Front Fixing Method: Numerical Analysis and Computing. Abstract and Applied Analysis, 2014, 1-9. doi:10.1155/2014/146745Griewank, P. J., & Notz, D. (2013). Insights into brine dynamics and sea ice desalination from a 1-D model study of gravity drainage. Journal of Geophysical Research: Oceans, 118(7), 3370-3386. doi:10.1002/jgrc.20247Javierre, E., Vuik, C., Vermolen, F. J., & van der Zwaag, S. (2006). A comparison of numerical models for one-dimensional Stefan problems. Journal of Computational and Applied Mathematics, 192(2), 445-459. doi:10.1016/j.cam.2005.04.062Briozzo, A. C., Natale, M. F., & Tarzia, D. A. (2007). Explicit solutions for a two-phase unidimensional LamĂ©âClapeyronâStefan problem with source terms in both phases. Journal of Mathematical Analysis and Applications, 329(1), 145-162. doi:10.1016/j.jmaa.2006.05.083Caldwell, J., & Chan, C.-C. (2000). Spherical solidification by the enthalpy method and the heat balance integral method. Applied Mathematical Modelling, 24(1), 45-53. doi:10.1016/s0307-904x(99)00031-1Chantasiriwan, S., Johansson, B. T., & Lesnic, D. (2009). The method of fundamental solutions for free surface Stefan problems. Engineering Analysis with Boundary Elements, 33(4), 529-538. doi:10.1016/j.enganabound.2008.08.010Hon, Y. C., & Li, M. (2008). A computational method for inverse free boundary determination problem. International Journal for Numerical Methods in Engineering, 73(9), 1291-1309. doi:10.1002/nme.2122RIZWAN-UDDIN. (1999). A Nodal Method for Phase Change Moving Boundary Problems. International Journal of Computational Fluid Dynamics, 11(3-4), 211-221. doi:10.1080/10618569908940875Caldwell, J., & Kwan, Y. Y. (2003). On the perturbation method for the Stefan problem with time-dependent boundary conditions. International Journal of Heat and Mass Transfer, 46(8), 1497-1501. doi:10.1016/s0017-9310(02)00415-5Stephan, K., & Holzknecht, B. (1976). Die asymptotischen lösungen fĂŒr vorgĂ€nge des erstarrens. International Journal of Heat and Mass Transfer, 19(6), 597-602. doi:10.1016/0017-9310(76)90042-9SavoviÄ, S., & Caldwell, J. (2003). Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. International Journal of Heat and Mass Transfer, 46(15), 2911-2916. doi:10.1016/s0017-9310(03)00050-4Kutluay, S., Bahadir, A. R., & ĂzdeĆ, A. (1997). The numerical solution of one-phase classical Stefan problem. Journal of Computational and Applied Mathematics, 81(1), 135-144. doi:10.1016/s0377-0427(97)00034-4Asaithambi, N. S. (1997). A variable time step Galerkin method for a one-dimensional Stefan problem. Applied Mathematics and Computation, 81(2-3), 189-200. doi:10.1016/0096-3003(95)00329-0Landau, H. G. (1950). Heat conduction in a melting solid. Quarterly of Applied Mathematics, 8(1), 81-94. doi:10.1090/qam/33441Churchill, S. W., & Gupta, J. P. (1977). Approximations for conduction with freezing or melting. International Journal of Heat and Mass Transfer, 20(11), 1251-1253. doi:10.1016/0017-9310(77)90134-xKutluay, S., & Esen, A. (2004). An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition. Applied Mathematics and Computation, 150(1), 59-67. doi:10.1016/s0096-3003(03)00197-8Esen, A., & Kutluay, S. (2004). A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method. Applied Mathematics and Computation, 148(2), 321-329. doi:10.1016/s0096-3003(02)00846-9Mitchell, S. L., & Vynnycky, M. (2016). On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. Journal of Computational and Applied Mathematics, 300, 259-274. doi:10.1016/j.cam.2015.12.021Meek, P. C., & Norbury, J. (1984). Nonlinear Moving Boundary Problems and a Keller Box Scheme. SIAM Journal on Numerical Analysis, 21(5), 883-893. doi:10.1137/0721057Tarzia, D. (2017). Relationship between Neumann solutions for two-phase LamĂ©-Clapeyron-Stefan problems with convective and temperature boundary conditions. Thermal Science, 21(1 Part A), 187-197. doi:10.2298/tsci140607003tPlemmons, R. J. (1977). M-matrix characterizations.Iânonsingular M-matrices. Linear Algebra and its Applications, 18(2), 175-188. doi:10.1016/0024-3795(77)90073-8Axelsson, O. (1994). Iterative Solution Methods. doi:10.1017/cbo978051162410
The impact of heat mitigation strategies on the energy balance of a neighborhood in Los Angeles
Heat mitigation strategies can reduce excess heat in urban environments. These strategies, including solar reflective cool roofs and pavements, green vegetative roofs, and street vegetation, alter the surface energy balance to reduce absorption of sunlight at the surface and subsequent transfer to the urban atmosphere. The impacts of heat mitigation strategies on meteorology have been investigated in past work at the mesoscale and global scale. For the first time, we focus on the effect of heat mitigation strategies on the surface energy balance at the neighborhood scale. The neighborhood under investigation is El Monte, located in the eastern Los Angeles basin in Southern California. Using a computational fluid dynamics model to simulate micrometeorology at high spatial resolution, we compare the surface energy balance of the neighborhood assuming current land cover to that with neighborhoodâwide deployment of green roof, cool roof, additional trees, and cool pavement as the four heat mitigation strategies. Of the four strategies, adoption of cool pavements led to the largest reductions in net radiation (downward positive) due to the direct impact of increasing pavement albedo on ground level solar absorption. Comparing the effect of each heat mitigation strategy shows that adoption of additional trees and cool pavements led to the largest spatialâmaximum air temperature reductions at 14:00h (1.0 and 2.0 °C, respectively). We also investigate how varying the spatial coverage area of heat mitigation strategies affects the neighborhoodâscale impacts on meteorology. Air temperature reductions appear linearly related to the spatial extent of heat mitigation strategy adoption at the spatial scales and baseline meteorology investigated here
One dimensional drift-diffusion between two absorbing boundaries: application to granular segregation
Motivated by a novel method for granular segregation, we analyze the one
dimensional drift-diffusion between two absorbing boundaries. The time
evolution of the probability distribution and the rate of absorption are given
by explicit formulae, the splitting probability and the mean first passage time
are also calculated. Applying the results we find optimal parameters for
segregating binary granular mixtures.Comment: RevTeX, 5 pages, 6 figure
Dissolution in a field
We study the dissolution of a solid by continuous injection of reactive
``acid'' particles at a single point, with the reactive particles undergoing
biased diffusion in the dissolved region. When acid encounters the substrate
material, both an acid particle and a unit of the material disappear. We find
that the lengths of the dissolved cavity parallel and perpendicular to the bias
grow as t^{2/(d+1)} and t^{1/(d+1)}, respectively, in d-dimensions, while the
number of reactive particles within the cavity grows as t^{2/(d+1)}. We also
obtain the exact density profile of the reactive particles and the relation
between this profile and the motion of the dissolution boundary. The extension
to variable acid strength is also discussed.Comment: 6 pages, 6 figures, 2-column format, for submission to PR
Super-hydrodynamic limit in interacting particle systems
This paper is a follow-up of the work initiated in [3], where it has been
investigated the hydrodynamic limit of symmetric independent random walkers
with birth at the origin and death at the rightmost occupied site. Here we
obtain two further results: first we characterize the stationary states on the
hydrodynamic time scale and show that they are given by a family of linear
macroscopic profiles whose parameters are determined by the current reservoirs
and the system mass. Then we prove the existence of a super-hyrdrodynamic time
scale, beyond the hydrodynamic one. On this larger time scale the system mass
fluctuates and correspondingly the macroscopic profile of the system randomly
moves within the family of linear profiles, with the randomness of a Brownian
motion.Comment: 22 page
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