The Floating Random Walk and Its Application to Monte Carlo Solutions of Heat Equations

Abstract

Previous article Next article The Floating Random Walk and Its Application to Monte Carlo Solutions of Heat EquationsA. Haji-Sheikh and E. M. SparrowA. Haji-Sheikh and E. M. Sparrowhttps://doi.org/10.1137/0114031PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] L. Rayleigh, On James Bernoulli's theorem in probabilities, Philos. Mag., 47 (1899), 246–251 CrossrefGoogle Scholar[2] H. Lewy, , R. Courant and , K. Friedrichs, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32–74 10.1007/BF01448839 MR1512478 CrossrefGoogle Scholar[3] John Todd, Experiments in the solution of differential equations by Monte Carlo methods, J. Washington Acad. Sci., 44 (1954), 377–381 MR0067584 0059.11101 Google Scholar[4] Louis W. Ehrlich, Monte Carlo solutions of boundary value problems involving the difference analogue of $\partial \sp{2}u/\partial x\sp{2} +\partial \sp{2}u/\partial y\sp{2}+Ky\sp{-1}\partial u/\partial y=0$, J. Assoc. Comput. Mach., 6 (1959), 204–218 MR0102163 0085.33503 CrossrefISIGoogle Scholar[5] John H. Curtiss, Sampling methods applied to differential and difference equations, Proceedings, Seminar on Scientific Computation, November, 1949, International Business Machines Corp., New York, N. Y., 1950, 87–109 MR0043564 Google Scholar[6] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, John Wiley, New York, 1964 0115.35308 Google Scholar[7] J. M. Hammersley and , D. C. Handscomb, Monte Carlo methods, Methuen & Co. Ltd., London, 1965vii+178 MR0223065 0121.35503 Google Scholar[8] S. Chandresekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys., 15 (1943), 1–89 10.1103/RevModPhys.15.1 MR0008130 0061.46403 CrossrefGoogle Scholar[9] Mervin E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist., 27 (1956), 569–589 MR0088786 0075.28902 CrossrefISIGoogle Scholar[10] A. Haji-Sheikh, Ph.D. Thesis, Application of Monte Carlo methods to thermal conduction problems, University of Minnesota, Minneapolis, 1965 Google Scholar[11] Wolfgang Wasow, On the mean duration of random walks, J. Research Nat. Bur. Standards, 46 (1951), 462–471 MR0047962 CrossrefISIGoogle Scholar[12] V. D. Barnett, The behavior of pseudo-random sequences generated on computers by the multiplicative congruential method, Math. Comp., 16 (1962), 63–69 MR0136046 0108.13701 CrossrefGoogle Scholar[13] B. W. Lindgren, Statistical theory, The Macmillan Co., New York, 1962xii+427 MR0143273 Google Scholar[14] E. M. Sparrow, Temperature distribution in an internally cooled heat-generating solid, J. Heat Transfer, 82 (1960), 389–392 CrossrefGoogle Scholar[15] H. S. Carslaw and , T. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Monte Carlo Method for Solving PDEMonte Carlo Methods for Partial Differential Equations With Applications to Electronic Design Automation | 3 September 2022 Cross Ref GPU-Friendly FRW Algorithms for Electrostatic Computation ProblemsMonte Carlo Methods for Partial Differential Equations With Applications to Electronic Design Automation | 3 September 2022 Cross Ref A Hybrid Random Walk Algorithm for 3-D Thermal AnalysisMonte Carlo Methods for Partial Differential Equations With Applications to Electronic Design Automation | 3 September 2022 Cross Ref Solving transient coupled conductive and radiative transfers in porous media with a Monte Carlo Method: Characterization of thermal conductivity of foams using a numerical Flash MethodInternational Journal of Thermal Sciences, Vol. 179 | 1 Sep 2022 Cross Ref Stochastic Simulation Algorithms for Solving Transient Anisotropic Diffusion-recombination Equations and Application to Cathodoluminescence ImagingMethodology and Computing in Applied Probability, Vol. 100 | 16 July 2022 Cross Ref Comprehensive correlation for the prediction of the heat transfer through a single droplet in dropwise condensation regimeApplied Thermal Engineering, Vol. 209 | 1 Jun 2022 Cross Ref A Global Random Walk on Spheres algorithm for calculating the solution and its derivatives of the drift‐diffusion‐reaction equationsMathematical Methods in the Applied Sciences, Vol. 45, No. 3 | 7 October 2021 Cross Ref Random walk on spheres algorithm for solving steady-state and transient diffusion-recombination problemsMonte Carlo Methods and Applications, Vol. 27, No. 4 | 4 November 2021 Cross Ref A global random walk on grid algorithm for second order elliptic equationsMonte Carlo Methods and Applications, Vol. 27, No. 4 | 27 October 2021 Cross Ref Estimating Permeability of Partially Frozen Soil Using Floating Random WalksWater Resources Research, Vol. 57, No. 12 | 7 December 2021 Cross Ref A Comprehensive Review on Multi-Dimensional Heat Conduction of Multi-Layer and Composite Structures: Analytical SolutionsJournal of Thermal Science, Vol. 30, No. 6 | 30 October 2021 Cross Ref A global random walk on grid algorithm for second order elliptic equationsMonte Carlo Methods and Applications, Vol. 27, No. 3 | 8 August 2021 Cross Ref A parallelizable mesh-free approach for simulation of heat conduction in ferromagnetic particulate materials under magnetic field effectInternational Journal of Heat and Mass Transfer, Vol. 174 | 1 Aug 2021 Cross Ref Random walk on ellipsoids method for solving elliptic and parabolic equationsMonte Carlo Methods and Applications, Vol. 26, No. 4 | 20 November 2020 Cross Ref Mesh-free stochastic algorithms for systems of drift–diffusion–reaction equations and anisotropic diffusion flux calculationsProbabilistic Engineering Mechanics, Vol. 61 | 1 Jul 2020 Cross Ref Effective floating volume: a highly parallelizable mesh-free approach for solving transient multiphysics problems in multi-scale geometries with non-linear material propertiesComputational Mechanics, Vol. 65, No. 3 | 21 November 2019 Cross Ref A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk methodNumerical Algorithms, Vol. 83, No. 2 | 11 April 2019 Cross Ref Stochastic algorithm for solving transient diffusion equations with a precise accounting of reflection boundary conditions on a substrate surfaceApplied Mathematics Letters, Vol. 96 | 1 Oct 2019 Cross Ref A random walk on small spheres method for solving transient anisotropic diffusion problemsMonte Carlo Methods and Applications, Vol. 25, No. 3 | 1 Sep 2019 Cross Ref A global random walk on spheres algorithm for transient heat equation and some extensionsMonte Carlo Methods and Applications, Vol. 25, No. 1 | 1 Mar 2019 Cross Ref First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equationsApplied Mathematics Letters, Vol. 88 | 1 Feb 2019 Cross Ref A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated DomainsDoklady Mathematics, Vol. 98, No. 2 | 10 November 2018 Cross Ref Application of the von Mises–Fisher distribution to Random Walk on Spheres method for solving high-dimensional diffusion–advection–reaction equationsStatistics & Probability Letters, Vol. 138 | 1 Jul 2018 Cross Ref Initial-boundary value problem for the heat equation—A stochastic algorithmThe Annals of Applied Probability, Vol. 28, No. 3 | 1 Jun 2018 Cross Ref A mesh-free Monte-Carlo method for simulation of three-dimensional transient heat conduction in a composite layered material with temperature dependent thermal propertiesInternational Journal of Heat and Mass Transfer, Vol. 119 | 1 Apr 2018 Cross Ref Random walk on spheres method for solving anisotropic drift-diffusion problemsMonte Carlo Methods and Applications, Vol. 24, No. 1 | 3 February 2018 Cross Ref Snowflake Melting Simulation Using Smoothed Particle HydrodynamicsJournal of Geophysical Research: Atmospheres, Vol. 123, No. 3 | 13 February 2018 Cross Ref Random walk on spheres method for solving anisotropic drift-diffusion problemsMonte Carlo Methods and Applications, Vol. 0, No. 0 | 3 Feb 2018 Cross Ref Monte Carlo Algorithms for Problems with Partially Reflecting BoundariesLarge-Scale Scientific Computing | 3 January 2018 Cross Ref The Spatial Diffusion of Social Conformity in the United States and Russia: The Case of Voting ParticipationSSRN Electronic Journal | 1 Jan 2018 Cross Ref Two-dimensional transient heat conduction in multi-layered composite media with temperature dependent thermal diffusivity using floating random walk Monte-Carlo methodInternational Journal of Heat and Mass Transfer, Vol. 115 | 1 Dec 2017 Cross Ref Monte Carlo Method Simulation for Two-Dimensional Heat Transfer in Homogenous Medium and Proposed Application to Quench Propagation SimulationIEEE Transactions on Applied Superconductivity, Vol. 27, No. 4 | 1 Jun 2017 Cross Ref Study on random walk and its application to solution of heat conduction equation by Monte Carlo methodProgress in Nuclear Energy, Vol. 96 | 1 Apr 2017 Cross Ref Walk-on-spheres algorithm for solving third boundary value problemApplied Mathematics Letters, Vol. 64 | 1 Feb 2017 Cross Ref Random walk on spheres algorithm for solving transient drift-diffusion-reaction problemsMonte Carlo Methods and Applications, Vol. 23, No. 3 | 1 Jan 2017 Cross Ref Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo MethodsMethods of Fourier Analysis and Approximation Theory | 12 March 2016 Cross Ref A hybrid random walk algorithm for 3-D thermal analysis of integrated circuits2014 19th Asia and South Pacific Design Automation Conference (ASP-DAC) | 1 Jan 2014 Cross Ref On the adoption of the Monte Carlo method to solve one-dimensional steady state thermal diffusion problems for non-uniform solidsApplied Mathematical Modelling, Vol. 37, No. 23 | 1 Dec 2013 Cross Ref Accelerated floating random walk algorithm for the electrostatic computation with 3-D rectilinear-shaped conductorsSimulation Modelling Practice and Theory, Vol. 34 | 1 May 2013 Cross Ref Diffusion and Geographical Equilibrium in Voter Turnout in the United States: 1920-2008SSRN Electronic Journal | 1 Jan 2012 Cross Ref Stochastic extraction for SoC and SiP interconnect with variability2011 IEEE Electrical Design of Advanced Packaging and Systems Symposium (EDAPS) | 1 Dec 2011 Cross Ref ReferencesFlow and Transport in Porous Media and Fractured Rock | 27 April 2011 Cross Ref Variation-aware stochastic extraction with large parameter dimensionality: Review and comparison of state of the art intrusive and non-intrusive techniques2011 12th International Symposium on Quality Electronic Design | 1 Mar 2011 Cross Ref The Spatial Diffusion of Voting Participation: Qualitative and Quantitative Predictions as Solutions to the Laplace EquationSSRN Electronic Journal | 1 Jan 2010 Cross Ref Floating Random WalkMonte Carlo Methods for Electromagnetics | 19 February 2010 Cross Ref Generalized Sommerfeld problem for time fractional diffusion equation: analytical and numerical approachJournal of Inverse and Ill-posed Problems, Vol. 17, No. 4 | 1 Jan 2009 Cross Ref Walk-on-Spheres Algorithm for Solving Boundary-Value Problems with Continuity Flux ConditionsMonte Carlo and Quasi-Monte Carlo Methods 2006 | 1 Jan 2008 Cross Ref Monte Carlo methods for solving elliptic equations with boundary conditions containing the normal derivativeDoklady Mathematics, Vol. 74, No. 2 | 1 Oct 2006 Cross Ref Particle-Tracking Random-Walk Computation of High-Peclet-Number ConvectionNumerical Heat Transfer, Part A: Applications, Vol. 50, No. 7 | 1 Aug 2006 Cross Ref Toward a more robust and accurate CEM fast Integral equation solver for IC applicationsIEEE Transactions on Advanced Packaging, Vol. 28, No. 3 | 1 Aug 2005 Cross Ref Monte Carlo Method for Transient Eddy-Current CalculationsIEEE Transactions on Magnetics, Vol. 40, No. 6 | 1 Nov 2004 Cross Ref Monte Carlo simulation of the effective thermal conductivity in two-phase materialJournal of Materials Processing Technology, Vol. 153-154 | 1 Nov 2004 Cross Ref Local Solutions in Potential Theory and Linear Elasticity Using Monte Carlo MethodsJournal of Applied Mechanics, Vol. 70, No. 3 | 11 June 2003 Cross Ref MONTE CARLO METHOD TO SOLVE MULTIDIMENSIONAL BIOHEAT TRANSFER PROBLEMNumerical Heat Transfer, Part B: Fundamentals, Vol. 42, No. 6 | 30 November 2010 Cross Ref Random walk method for the two‐ and three‐dimensional Laplace, Poisson and Helmholtz's equationsInternational Journal for Numerical Methods in Engineering, Vol. 51, No. 10 | 10 Aug 2001 Cross Ref Using computational fluid dynamics in probabilistic engineering design15th AIAA Computational Fluid Dynamics Conference | 22 August 2012 Cross Ref A Monte Carlo solution method for linear elasticityInternational Journal of Solids and Structures, Vol. 37, No. 42 | 1 Oct 2000 Cross Ref Monte Carlo MethodsNumerical Techniques in Electromagnetics, Second Edition | 13 January 2010 Cross Ref Thermal studies of a superconducting current limiter using Monte-Carlo methodThe European Physical Journal Applied Physics, Vol. 7, No. 1 | 15 July 1999 Cross Ref Superconducting resin banded coil: a modified convection coefficientCryogenics, Vol. 39, No. 1 | 1 Jan 1999 Cross Ref A monte carlo approach for simulation of heat flow in sand and metal mold castings (virtual mold modeling)Metallurgical and Materials Transactions B, Vol. 29, No. 2 | 1 Apr 1998 Cross Ref The stochastic modeling of solidification structures in alloy 718 remelt ingotsJOM, Vol. 50, No. 3 | 1 Mar 1998 Cross Ref Stochastic iterative methods for solving parabolic equationsSiberian Mathematical Journal, Vol. 38, No. 5 | 1 Oct 1997 Cross Ref Whole field computation using Monte Carlo methodInternational Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 10, No. 5 | 1 Sep 1997 Cross Ref Whole field computation using Monte Carlo methodInternational Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 10, No. 5 | 1 Sep 1997 Cross Ref Whole field computation using Monte Carlo methodInternational Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 10, No. 5 | 1 Sep 1997 Cross Ref Simulation of diffusion in 2-D heterogeneous systems: comparison with effective medium and percolation theoriesPhysica A: Statistical Mechanics and its Applications, Vol. 227, No. 3-4 | 1 Jun 1996 Cross Ref A triangular mesh random walk for Dirichlet problemsJournal of the Franklin Institute, Vol. 332, No. 5 | 1 Sep 1995 Cross Ref A BOUNDARY-DISPATCH MONTE CARLO (EXODUS) METHOD FOR ANALYSIS OF CONDUCTIVE HEAT TRANSFER PROBLEMSNumerical Heat Transfer, Part B: Fundamentals, Vol. 24, No. 4 | 1 Dec 1993 Cross Ref Monte Carlo solution of axisymmetric potential problemsIEEE Transactions on Industry Applications, Vol. 29, No. 6 | 1 Nov 1993 Cross Ref Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealingReviews of Modern Physics, Vol. 65, No. 4 | 1 October 1993 Cross Ref Monte Carlo methods in an introductory electromagnetic courseIEEE Transactions on Education, Vol. 33, No. 1 | 1 Feb 1990 Cross Ref A bibliography on applications of random walks in theoretical chemistry and physicsAdvances in Molecular Relaxation and Interaction Processes, Vol. 22, No. 1 | 1 Jan 1982 Cross Ref The Monte Carlo methodJournal of Soviet Mathematics, Vol. 7, No. 2 | 1 Feb 1977 Cross Ref A modification of the Monte Carlo method for steady heat conduction problemsInternational Journal for Numerical Methods in Engineering, Vol. 10, No. 5 | 1 Jan 1976 Cross Ref A method of simulating the crust formation in the ore bath of smelting furnacesJournal of Engineering Physics, Vol. 26, No. 6 | 1 Jun 1974 Cross Ref Monte Carlo simulation of simultaneous bulk, grain boundary, and surface diffusionThe Journal of Chemical Physics, Vol. 59, No. 10 | 15 Nov 1973 Cross Ref On mathematical analysis of gas transport in the lungRespiration Physiology, Vol. 18, No. 3 | 1 Sep 1973 Cross Ref The conductivity of a medium interrupted by voidsZeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 22, No. 3 | 1 May 1971 Cross Ref Solution of partial differential equations by a modified random walkNumerische Mathematik, Vol. 18, No. 1 | 1 Feb 1971 Cross Ref Monte Carlo solution of axisymmetric potential problemsConference Record of the 1990 IEEE Industry Applications Society Annual Meeting Cross Ref Probabilistic (Monte Carlo) Modeling of Solidification StructureModeling and Simulation of Microstructure Evolution in Solidifying Alloys Cross Ref Solution of one mixed problem for equation of relaxational filtration by Monte Carlo methodsAdvances in High Performance Computing and Computational Sciences Cross Ref Random Walks for Solving Boundary-Value Problems with Flux ConditionsNumerical Methods and Applications Cross Ref Volume 14, Issue 2| 1966SIAM Journal on Applied Mathematics209-416 History Submitted:14 May 1965Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114031Article page range:pp. 370-389ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics