Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

JOURNAL OF COMPUTATIONAL PHYSICS

卷:379

Article

Grebenkov, Denis S.^1,2  Traytak, Sergey D.³ 

[1] Univ Paris Saclay, Ecole Polytech, CNRS, Lab Phys Matiere Condensee UMR 7643, F-91128 Palaiseau, France

[2] CNRS, Interdisciplinary Sci Ctr Poncelet, UMI IUM IITP RAS Steklov MI RAS Skoltec HSE 2615, Bolshoy Vlasyevskiy Pereulok 11, Moscow 119002, Russia

[3] Russian Acad Sci, Semenov Inst Chem Phys, 4 Kosygina St, Moscow 117977, Russia

关键词: Green function;    Laplace operator;    Boundary value problem;    Diffusion-reaction;    Semi-analytical solution;   

DOI  :  10.1016/j.jcp.2018.10.033

来源: Elsevier

【 摘 要 】

The generalized method of separation of variables (GMSV) is applied to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (e.g., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and solution of a system of linear algebraic equations, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free