×

zbMATH — the first resource for mathematics

Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle. (English) Zbl 1404.65194
Summary: The goal of this paper is to develop a highly accurate and efficient numerical method for the solution of a time-dependent partial differential equation with a piecewise constant coefficient, on a finite interval with periodic boundary conditions. The resulting algorithm can be used, for example, to model the diffusion of heat energy in one space dimension, in the case where the spatial domain represents a medium consisting of two homogeneous materials. The resulting model has, to our knowledge, not yet been solved in closed form through analytical methods, and is difficult to solve using existing numerical methods, thus suggesting an alternative approach. The approach presented in this paper is to represent the solution as a linear combination of wave functions that change frequencies at the interfaces between different materials. It is demonstrated through numerical experiments that using the Uncertainty Principle to construct a basis of such functions, in conjunction with a spectral method, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with conventional time-stepping methods.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35K05 Heat equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amrein, WO; Berthier, AM, On support properties of \(l^p\)-functions and their Fourier transforms, J Funct Anal, 24, 258-267, (1977) · Zbl 0355.42015
[2] Arfken GB, Weber HJ, Harris F (2012) Mathematical methods for physicists: a comprehensive guide, 7th edn. Academic Press, Cambridge · Zbl 1239.00005
[3] Aronson, DG, Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 22, 607-694, (1968) · Zbl 0182.13802
[4] Aurko A, Lambers JV (2017) Computing eigenvalues of 2-d differential operators with piecewise constant coefficients. In preparation
[5] Benedicks, M, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J Math Anal Appl, 106, 180-183, (1985) · Zbl 0576.42016
[6] Burden RL, Faires JD (2010) Numerical analysis, 9th edn. Cengage Learning, Bostan · Zbl 0671.65001
[7] Carleman T (1960) Complete works. Malmo Litös, Malmo · Zbl 0098.00104
[8] Étoré, P, On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electron J Probab, 11, 249-275, (2006) · Zbl 1112.60061
[9] Evans LC (1994) Partial differential equations, 2nd edn. Graduate studies in mathematics. American Mathematical Society, New York
[10] Farlow SJ (1993) Partial differential equations for scientists and engineers. Dover Publications, Inc., New York · Zbl 0851.35001
[11] Fefferman, C, The uncertainty principle, Bull Am Math Soc, 9, 129-206, (1983) · Zbl 0526.35080
[12] Filoche, M; Mayaboroda, S; Patterson, B, Localization of eigenfunctions of a one-dimensional elliptic operator, Contemp Math, 581, 99-116, (2012) · Zbl 1321.35116
[13] Folland GB (1995) Introduction to partial differential equations, 2nd edn. Princeton University Press, Princeton · Zbl 0841.35001
[14] Gerardo-Giorda, L; Tallec, P; Nataf, F, A Robin-Robin preconditioner for advection-diffusion equations with discontinuous coefficients, Comput Methods Appl Mech Eng, 193, 745-764, (2004) · Zbl 1053.76039
[15] Golub GH, van Loan C (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore · Zbl 0865.65009
[16] Guidotti, P; Lambers, JV; Sølna, K, Analysis of the 1d wave equation in inhomogeneous media, Numer Funct Anal Optim, 27, 25-55, (2006) · Zbl 1096.35081
[17] Gustafsson B, Kreiss HO, Oliger J (1995) Time dependent problems and difference methods. Wiley, Amsterdam · Zbl 0843.65061
[18] Heisenberg, W, Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik, Zeitschrift für Physik, 43, 172-198, (1927) · JFM 53.0853.05
[19] Hoteit, H; Mose, R; Younes, A; Lehmann, F; Ackerer, P, Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods, Math Geol, 34, 435-456, (2002) · Zbl 1107.76401
[20] LaBolle, EM; Fogg, GE; Tompson, AFB, Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods, Water Resour Res, 32, 583-593, (1996)
[21] Ladyženskaja OA, Solonnikov VA, Ural’ceva NN (1968) Linear and quasi-linear equations of parabolic type. In: Translations of mathematical monographs, vol 23. American Mathematical Society, Providence, RI, p 648
[22] Lambers, JV, Approximate diagonalization of variable-coefficient differential operators through similarity transformations, Comput Math Appl, 64, 2575-2593, (2012) · Zbl 1268.65101
[23] Lejay, A; Pichot, G, Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps, J Comput Phys, 231, 7299-7314, (2012) · Zbl 1284.65007
[24] Long SD, Sheikholeslami S, Lambers JV (2017) Diagonalization of 1-d differential operators with piecewise constant coefficients. In preparation
[25] Portenko NI (1990) Generalized diffusion processes. In: Translations of mathematical monographs (trans: McFaden HH), vol 83. American Mathematical Society (AMS), Providence, RI, p 180
[26] Shampine, LF; Reichelt, MW, The Matlab ODE suite, SIAM J Sci Comp, 18, 1-22, (1997) · Zbl 0868.65040
[27] Uffink, GJM, A random walk method for the simulation of macrodispersion in a stratified aquifer, Relat Groundw Quantity Q, 146, 103-114, (1985)
[28] Walsh, JB, A diffusion with discontinuous local time, Temps Locaux, 52-53, 37-45, (1978)
[29] Weyl, H, Ramifications, old and new, of the eigenvalue problem, Bull Am Math Soc, 56, 115-139, (1950) · Zbl 0041.21003
[30] Zunino P (2003) Iterative substructuring methods for advection-diffusion problems in heterogeneous media. In: Eberhard Bänsch (ed) Challenges in scientific computing—CISC 2002, volume 35 of Lecture Notes in Computational Science and Engineering, Springer, New York, pp 184-210 · Zbl 1043.65115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.