A random walk on small spheres method for solving transient
anisotropic diffusion problems

Irina Shalimova; Karl K. Sabelfeld

doi:10.1515/mcma-2019-2047

Article

A random walk on small spheres method for solving transient anisotropic diffusion problems

Published/Copyright: August 20, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Monte Carlo Methods and Applications Volume 25 Issue 3

Abstract

A meshless stochastic algorithm for solving anisotropic transient diffusion problems based on an extension of the classical Random Walk on Spheres method is developed. Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have derived approximations of the probability densities for the first passage time and the exit point on a small sphere. The method can be conveniently applied to solve diffusion problems with spatially varying diffusion coefficients and is simply implemented for complicated three-dimensional domains. Particle tracking algorithm is highly efficient for calculation of fluxes to boundaries. We present some simulation results in the case of cathodoluminescence and electron beam induced current in the vicinity of a dislocation in a semiconductor material.

Keywords: Anisotropic diffusion equation; spherical mean value relation; flux

MSC 2010: 65C05; 65C40; 65Z05

Funding source: Russian Science Foundation

Award Identifier / Grant number: 19-11-00019

Funding statement: Support of the Russian Science Foundation under Grant 19-11-00019 is gratefully acknowledged.

A The reciprocity relation

In this section we deal with the pair of equations, the direct (3.1) and adjoint (3.2), in the domain G with Dirichlet boundary conditions wd⁢(𝐲,τ)=0, 𝐲∈Γ=∂⁡G and wa⁢(𝐲,τ¯)=1 if 𝐲∈Γ1 and wa⁢(𝐲,τ¯)=0 if 𝐲∈Γ2; the initial condition is zero for both solutions. For simplicity we consider here the case when Γ=Γ1∪Γ2, that is easily extended to any general case.

Let us multiply the direct equation (3.1) by wa⁢(𝐱,τ¯)

wa⁢(𝐱,τ¯)⁢∂∂⁡τ⁢wd⁢(𝐱,τ)=wa⁢(𝐱,τ¯)⁢Δd⁢wd⁢(𝐱,τ)+wa⁢(𝐱,τ¯)⁢δ⁢(𝐱-𝐱0)⁢δ⁢(τ)

and the adjoint equation (3.2) by wd⁢(𝐱,τ)

wd⁢(𝐱,τ)⁢∂∂⁡τ¯⁢wa⁢(𝐱,τ¯)=wd⁢(𝐱,τ)⁢Δd⁢wa⁢(𝐱,τ¯).

Then subtract one equation from another one and integrate the result over the volume G and time interval [0,t]. This yields

0=∫0t∫G(wa⁢(𝐱,t-τ)⁢Δd⁢wd⁢(𝐱,τ)-wd⁢(𝐱,τ)⁢Δd⁢wa⁢(𝐱,t-τ))⁢𝑑Vx⁢𝑑τ+wa⁢(𝐱0,t),

where we make use of τ¯=t-τ and zero initial condition for both of function wd⁢(𝐱,τ) and wa⁢(𝐱,t-τ). The volume integral is transformed to surface integrals by the Gauss formula [28]

0=∫0t∫Γ1+Γ2{wa⁢∑i=13di⁢∂⁡wd∂⁡xi⁢cos⁡γi-wd⁢∑i=13di⁢∂⁡wa∂⁡xi⁢cos⁡γi}⁢𝑑σ⁢𝑑τ+wa⁢(𝐱0,t),

where γi are the angles between the directions xi (i=1,2,3), and the unit vector directed normally outward from the surface. Due to the boundary conditions for wd and wa, here the only nonzero term is wa on Γ1, which is given by wa⁢(𝐲,t-τ)=1 on 𝐲∈Γ1. From this we get

wa⁢(𝐱0,t)=-∫0t∫Γ1∑i=13di⁢∂⁡wd∂⁡xi⁢cos⁡γi⁢d⁢σ⁢d⁢τ,

that is a flux of particles to the boundary Γ1.

References

[1] R. Balescu, Transport Processes in Plasmas: Neoclassical Transport Theory. Transport Processes in Plasmas, North-Holland, Amsterdam, 1988. 10.1016/B978-0-444-87091-9.50009-9Search in Google Scholar

[2] R. L. Bell and C. A. Hogarth, Anisotropic diffusion lengths in germanium and silicon crystals containing parallel arrays of edge dislocations, J. Electronics Control 3 (1957), no. 5, 455–470. 10.1080/00207215708937106Search in Google Scholar

[3] G. Dagan, Flow and Transport in Porous Formations, Springer, Berlin, 1989. 10.1007/978-3-642-75015-1Search in Google Scholar

[4] L. Devroye, The series method for random variate generation and its application to the Kolmogorov–Smirnov distribution, Amer. J. Math. Management Sci. 1 (1981), no. 4, 359–379. 10.1080/01966324.1981.10737080Search in Google Scholar

[5] F. A. Dorini and M. C. C. Cunha, On the linear advection equation subject to random velocity fields, Math. Comput. Simulation 82 (2011), no. 4, 679–690. 10.1016/j.matcom.2011.10.008Search in Google Scholar

[6] C. Fleming, M. Mascagni and N. Simonov, An efficient Monte Carlo approach for solving linear problems in biomolecular electrostatics, Lecture Notes in Comput. Sci. 3516, Springer, Heidelberg (2005), 760–765. 10.1007/11428862_103Search in Google Scholar

[7] C. W. Gardiner, Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences, 2nd ed., Springer Ser. Synergetics 13, Springer, Berlin, 1985. 10.1007/978-3-662-02452-2Search in Google Scholar

[8] J. A. Given, J. B. Hubbard and J. F. Douglas, A first-passage algorithm for the hydrodynamic friction and diffusion-limited reaction rate of macromolecules, J. Chem. Phys. 106 (1997), no. 9, 3761–3771. 10.1063/1.473428Search in Google Scholar

[9] A. Haji-Sheikh and E. M. Sparrow, The floating random walk and its application to Monte Carlo solutions of heat equations, SIAM J. Appl. Math. 14 (1966), 370–389. 10.1137/0114031Search in Google Scholar

[10] O. A. Kurbanmuradov and K. K. Sabelfeld, Agrangian stochastic models for turbulent dispersion in the atmospheric boundary layer, Boundary-Layer Meteorol. 97 (2000), no. 2, 191–218. 10.1023/A:1002701813926Search in Google Scholar

[11] O. A. Kurbanmuradov, K. K. Sabelfeld, O. F. Smidts and H. Vereecken, A Lagrangian stochastic model for the transport in statistically homogeneous porous media, Monte Carlo Methods Appl. 9 (2003), no. 4, 341–366. 10.1515/156939603322601969Search in Google Scholar

[12] T. Lagache and D. Holcman, Extended narrow escape with many windows for analyzing viral entry into the cell nucleus, J. Stat. Phys. 166 (2017), no. 2, 244–266. 10.1007/s10955-016-1691-9Search in Google Scholar

[13] L. L. Latour, P. P. Mitra, R. L. Kleinberg and C. H. Sotak, Time-dependent diffusion coefficient of fluids in porous media as a probe of surface-to-volume ratio, J. Magnetic Resonance Ser. A 101 (1993), no. 3, 342–346. 10.1006/jmra.1993.1056Search in Google Scholar

[14] W. Liu, J. F. Carlin, N. Grandjean, B. Deveaud and G. Jacopin, Exciton dynamics at a single dislocation in GaN probed by picosecond time-resolved cathodoluminescence, Applied Physics Letters 109 (2016), no. 4, Article ID 042101. 10.1063/1.4959832Search in Google Scholar

[15] Y. Lou, Diffusion, Self-Diffusion and Cross-Diffusion, ProQuest LLC, Ann Arbor, 1995; Ph.D. thesis, University of Minnesota, 1995. Search in Google Scholar

[16] M. Mascagni and N. A. Simonov, Monte Carlo methods for calculating some physical properties of large molecules, SIAM J. Sci. Comput. 26 (2004), no. 1, 339–357. 10.1137/S1064827503422221Search in Google Scholar

[17] M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist. 27 (1956), 569–589. 10.1214/aoms/1177728169Search in Google Scholar

[18] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, 2002. 10.1201/9781420035322Search in Google Scholar

[19] K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer Ser. Comput. Phys., Springer, Berlin, 1991. 10.1007/978-3-642-75977-2Search in Google Scholar

[20] K. K. Sabelfeld, Random walk on spheres method for solving drift-diffusion problems, Monte Carlo Methods Appl. 22 (2016), no. 4, 265–275. 10.1515/mcma-2016-0118Search in Google Scholar

[21] K. K. Sabelfeld, A mesh free floating random walk method for solving diffusion imaging problems, Statist. Probab. Lett. 121 (2017), 6–11. 10.1016/j.spl.2016.10.006Search in Google Scholar

[22] K. K. Sabelfeld, Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems, Monte Carlo Methods Appl. 23 (2017), no. 3, 189–212. 10.1515/mcma-2017-0113Search in Google Scholar

[23] K. K. Sabelfeld, V. Kaganer, C. Pfüller and O. Brandt, Dislocation contrast in cathodoluminescence and electron-beam induced current maps on GaN(0001), J. Phys. D 50 (2017), Article ID 405101. 10.1088/1361-6463/aa85c8Search in Google Scholar

[24] K. K. Sabelfeld and I. A. Shalimova, Spherical and Plane Integral Operators for PDEs. Construction, Analysis, and Applications, De Gruyter, Berlin, 2013. 10.1515/9783110315332Search in Google Scholar

[25] I. Shalimova and K. K. Sabelfeld, Random walk on spheres method for solving anisotropic drift-diffusion problems, Monte Carlo Methods Appl. 24 (2018), no. 1, 43–54. 10.1515/mcma-2018-0006Search in Google Scholar

[26] N. A. Simonov, M. Mascagni and M. O. Fenley, Monte Carlo-based linear Poisson–Boltzmann approach makes accurate salt-dependent solvation free energy predictions possible, J. Chem. Phys. 127 (2007), Article ID 185105. 10.1063/1.2803189Search in Google Scholar PubMed

[27] A. Singer, Z. Schuss and D. Holcman, Narrow escape II: The circular disk, J. Stat. Phys. 122 (2006), no. 3, 465–489. 10.1007/s10955-005-8027-5Search in Google Scholar

[28] V. I. Smirnov, A Course of Higher Mathematics. Vol. IV. Part Two, Pergamon Press, Oxford, 1964. Search in Google Scholar

[29] T. Vesala, U. Rannik, M. Leclerc, T. Foken and K. K. Sabelfeld, Flux and concentration footprints, Agricult. Forest Meteorol. 127 (2004), no. 3–4, 111–116. 10.1016/j.agrformet.2004.07.007Search in Google Scholar

[30] Q. Zhang, A multi-length-scale theory of the anomalous mixing-length growth for tracer flow in heterogeneous porous media, J. Stat. Phys. 66 (1992), no. 1–2, 485–501. 10.1007/BF01060076Search in Google Scholar

[31] Z. Zhang, D. L. Johnson and L. M. Schwartz, Simulating the time-dependent diffusion coefficient in mixed-poresize materials, Phys. Rev. E 84 (2011), Article ID 031129. 10.1103/PhysRevE.84.031129Search in Google Scholar PubMed

Received: 2019-05-07

Revised: 2019-08-12

Accepted: 2019-08-14

Published Online: 2019-08-20

Published in Print: 2019-09-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/mcma-2019-2047

Keywords for this article

Anisotropic diffusion equation; spherical mean value relation; flux