Analytic expressions of the solutions of advection-diffusion problems in one dimension with discontinuous coefficients. (English) Zbl 1495.65014
Summary: In this article, we provide a method to compute analytic expressions of the resolvent kernel of differential operators of the diffusion type with discontinuous coefficients in one dimension. Then we apply it when the coefficients are piecewise constant. We also perform the Laplace inversion of the resolvent kernel to obtain expressions of the transition density functions or fundamental solutions. We show how these explicit formula are useful to simulate advection-diffusion problems using particle tracking techniques.
MSC:
| 65C35 | Stochastic particle methods |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K57 | Reaction-diffusion equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
Keywords:
advection-diffusion problems; resolvent kernels; transition density functions; discontinuous coefficients; particle tracking techniquesReferences:
| [1] | M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th ed., Dover, New York, 1970. · Zbl 0171.38503 |
| [2] | T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, and B. Wood, Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Ann. Appl. Probab., 21 (2011), pp. 183-214, https://doi.org/10.1214/10-AAP691. · Zbl 1226.60113 |
| [3] | T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, and B. Wood, Corrections: Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Ann. Appl. Probab., 21 (2011), pp. 2050-2051, https://doi.org/10.1214/11-AAP775. · Zbl 1226.60113 |
| [4] | D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. (N.S.), 73 (1967), pp. 890-896, https://doi.org/10.1090/S0002-9904-1967-11830-5. · Zbl 0153.42002 |
| [5] | M. Bechtold, J. Vanderborght, O. Ippisch, and H. Vereecken, Efficient random walk particle tracking algorithm for advective-dispersive transport in media with discontinuous dispersion coefficients and water contents, Water Res. Res., 47 (2011), W10526, https://doi.org/10.1029/2010WR010267. |
| [6] | S. Berezin and O. Zayats, Skew Brownian Motion with dry friction: The Pugachev-Sveshnikov equation approach, Mater. Phys. Mech., 41 (2019), pp. 103-110. |
| [7] | A. N. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, 2nd ed., Probab. Appl., Birkhäuser, Basel, 2002. · Zbl 1012.60003 |
| [8] | M. Bossy, N. Champagnat, S. Maire, and D. Talay, Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics, ESAIM Math. Model. Numer. Anal., 44 (2010), pp. 997-1048, https://doi.org/10.1051/m2an/2010050. · Zbl 1204.82020 |
| [9] | R. Cantrell and C. Cosner, Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design, Theor. Popul. Biol., 55 (1999), pp. 189-207, https://doi.org/10.1006/tpbi.1998.1397. · Zbl 0958.92028 |
| [10] | Z.-Q. Chen and M. Zili, One-dimensional heat equation with discontinuous conductance, Sci. China Math., 58 (2015), pp. 97-108, https://doi.org/10.1007/s11425-014-4912-1. · Zbl 1322.60129 |
| [11] | M. Decamps, A. De Schepper, and M. Goovaerts, Applications of \(\delta \)-function perturbation to the pricing of derivative securities, Phys. A, 342 (2004), pp. 677-692, https://doi.org/10.1016/j.physa.2004.05.035. |
| [12] | F. Delay, P. Ackerer, and C. Danquigny, Simulating solute transport in porous or fractured formations using random walks particle tracking: A review, Vadose Zone J., 4 (2005), pp. 360-379. |
| [13] | D. Dereudre, S. Mazzonetto, and S. Roelly, An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers, Monte Carlo Methods Appl., 22 (2016), pp. 1-23, https://doi.org/10.1515/mcma-2016-0100. · Zbl 1335.60151 |
| [14] | D. Dereudre, S. Mazzonetto, and S. Roelly, Exact simulation of Brownian diffusions with drift admitting jumps, SIAM J. Sci. Comput., 39 (2017), pp. A711-A740, https://doi.org/10.1137/16M107699X. · Zbl 1370.60113 |
| [15] | J. Eckhardt and G. Teschl, Sturm-Liouville operators with measure-valued coefficients, J. Anal. Math., 120 (2013), pp. 151-224. · Zbl 1315.34001 |
| [16] | P. Étoré, On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electron. J. Probab., 11 (2006), pp. 249-275, https://doi.org/10.1214/EJP.v11-311. · Zbl 1112.60061 |
| [17] | P. Étoré and A. Lejay, A Donsker theorem to simulate one-dimensional processes with measurable coefficients, ESAIM Probab. Stat., 11 (2007), pp. 301-326, https://doi.org/10.1051/ps:2007021. · Zbl 1181.60123 |
| [18] | P. Étoré and M. Martinez, Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process, Monte Carlo Methods Appl., 19 (2013), pp. 41-71. · Zbl 1269.65007 |
| [19] | W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), pp. 1-31. · Zbl 0059.11601 |
| [20] | W. Feller, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. (2), 60 (1954), pp. 417-436. · Zbl 0057.09805 |
| [21] | W. Feller, On second order differential operators, Ann. of Math. (2), 61 (1955), pp. 90-105. · Zbl 0064.11301 |
| [22] | W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math., 1 (1957), pp. 459-504. · Zbl 0077.29102 |
| [23] | W. Feller, On the intrinsic form for second order differential operator, Illinois J. Math., 2 (1959), pp. 1-18. · Zbl 0078.07601 |
| [24] | E. R. Fernholz, T. Ichiba, and I. Karatzas, Two Brownian particles with rank-based characteristics and skew-elastic collisions, Stochastic Process. Appl., 123 (2013), pp. 2999-3026. · Zbl 1296.60148 |
| [25] | A. Gairat and V. Shcherbakov, Density of skew Brownian motion and its functionals with application in finance, Math. Finance, 26 (2016), pp. 1069-1088, https://doi.org/10.1111/mafi.12120. · Zbl 1411.91555 |
| [26] | E. M. Garon and J. V. Lambers, Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle, Comput. Appl. Math., 37 (2018), pp. 2566-2587, https://doi.org/10.1007/s40314-017-0465-6. · Zbl 1404.65194 |
| [27] | B. Gaveau, M. Okada, and T. Okada, Second order differential operators and Dirichlet integrals with singular coefficients, I. Functional calculus of one-dimensional operators, Tohoku Math. J. (2), 39 (1987), pp. 465-504. · Zbl 0653.35034 |
| [28] | U. Gräwe, E. Deleersnijder, S. H. A. M. Shah, and A. W. Heemink, Why the Euler scheme in particle tracking is not enough: The shallow-sea pycnocline test case, Ocean Dynam., 62 (2012), pp. 501-514, https://doi.org/10.1007/s10236-012-0523-y. |
| [29] | Y. Güldü, On discontinuous Dirac operator with eigenparameter dependent boundary and two transmission conditions, Bound. Value Probl. (2016), 135, https://doi.org/10.1186/s13661-016-0639-y. · Zbl 1342.34027 |
| [30] | H. Hoteit, R. Mose, A. Younes, F. Lehmann, and P. Ackerer, Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods, Math. Geol., 34 (2002), pp. 435-456. · Zbl 1107.76401 |
| [31] | K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, 2nd ed., Springer, Berlin, 1974. · Zbl 0285.60063 |
| [32] | E. M. LaBolle, G. E. Fogg, and A. F. B. Tompson, Random-walk simulation of transport in heterogeneous porous media: Local mass-conservation problem and implementation methods, Water Res. Res., 32 (1996), pp. 583-593, https://doi.org/10.1029/95WR03528. |
| [33] | O. A. Ladyženskaja, V. J. Rivkind, and N. N. Ural‘ceva, Equations aux dérivées partielles de type elliptique, Monogr. Univ. Math. 31, Dunod, Paris, 1968. · Zbl 0164.13001 |
| [34] | O. A. Ladyženskaja, V. J. Rivkind, and N. N. Ural‘ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 33, American Mathematical Society, Providence, RI, 1968. · Zbl 0174.15403 |
| [35] | O. A. Ladyženskaja, V. J. Rivkind, and N. N. Ural‘ceva, Classical solvability of diffraction problems for equations of elliptic and parabolic types, Dokl. Akad. Nauk SSSR, 158 (1964), pp. 513-515. |
| [36] | J. Langebrake, L. Riotte-Lambert, C. W. Osenberg, and P. De Leenheer, Differential movement and movement bias models for marine protected areas, J. Math. Biol., 64 (2012), pp. 667-696. · Zbl 1262.34049 |
| [37] | H. Langer and W. Schenk, Knotting of one-dimensional Feller processes, Math. Nachr., 113 (1983), pp. 151-161. · Zbl 0532.60066 |
| [38] | J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process, in Stochastic Analysis, Lecture Notes in Math. 1095, Springer, Berlin, 1984, pp. 51-82. · Zbl 0551.60059 |
| [39] | A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv., 3 (2006), pp. 413-466. · Zbl 1189.60145 |
| [40] | A. Lejay, Simulation of a stochastic process in a discontinuous layered medium, Electron. Commun. Probab., 16 (2011), pp. 764-774. · Zbl 1243.60062 |
| [41] | A. Lejay, L. Lenôtre, and G. Pichot, An exponential timestepping algorithm for diffusion with discontinuous coefficients, J. Comput. Phys., 396 (2019), pp. 888-904. · Zbl 1452.65008 |
| [42] | A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients, Ann. Appl. Probab., 16 (2006), pp. 107-139. · Zbl 1094.60056 |
| [43] | A. Lejay and G. Pichot, Simulating diffusion processes in discontinuous media: A numerical scheme with constant time steps, J. Comput. Phys., 231 (2012), pp. 7299-7314. · Zbl 1284.65007 |
| [44] | A. Lejay and G. Pichot, Simulating diffusion processes in discontinuous media: Benchmark tests, J. Comput. Phys., 314 (2016), pp. 384-413, https://doi.org/10.1016/j.jcp.2016.03.003. · Zbl 1349.65019 |
| [45] | F. T. Lindstrom and F. Oberhettinger, A note on a Laplace transform pair associated with mass transport in porous media and heat transport problems, SIAM J. Appl. Math., 29 (1975), pp. 288-292. · Zbl 0325.44001 |
| [46] | A. Lipton, Oscillating Bachelier and Black-Scholes formulas, World Scientific, in Financial Engineering, Hackensack, NJ, 2018, pp. 371-394, https://doi.org/10.1142/10425. |
| [47] | M. Martinez, Interprétations probabilistes d’opérateurs sous forme divergence et analyse de méthodes numériques probabilistes associées, Ph.D. thesis, Université de Provence, Marseille, France, 2004. |
| [48] | H. P. McKean, Jr., Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc., 82 (1956), pp. 519-548, https://doi.org/10.1090/S0002-9947-1956-0087012-3. · Zbl 0070.32003 |
| [49] | T. Okada, Asymptotic behavior of skew conditional heat kernels on graph networks, Canad. J. Math., 45 (1993), pp. 863-878, https://doi.org/10.4153/CJM-1993-049-6. · Zbl 0799.58084 |
| [50] | J. Pitman and M. Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches, Bernoulli, 9 (2003), pp. 1-24, https://doi.org/10.3150/bj/1068129008. · Zbl 1024.60032 |
| [51] | P. Puri and P. Kythe, Some inverse Laplace transforms of exponential form, Z. Angew. Math. Phys., 39 (1988), pp. 150-156, https://doi.org/10.1007/BF00945761. · Zbl 0644.44002 |
| [52] | J. M. Ramirez, E. A. Thomann, E. C. Waymire, J. Chastanet, and B. D. Wood, A note on the theoretical foundations of particle tracking methods in heterogeneous porous media, Water Resour. Res., 44 (2008), W01501, https://doi.org/10.1029/2007WR005914. |
| [53] | P. Salamon, D. Fernàndez-Garcia, and J. J. Gómez-Hernández, A review and numerical assessment of the random walk particle tracking method, J. Contam. Hydrol., 87 (2006), pp. 277-305, https://doi.org/10.1016/j.jconhyd.2006.05.005. |
| [54] | D. Spivakovskaya, A. W. Heemink, and E. Deleersnijder, Lagrangian modelling of multi-dimensional advection-diffusion with space-varying diffusivities: Theory and idealized test cases, Ocean Dynam., 57 (2007), pp. 189-203. |
| [55] | D. W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, in Séminaire de Probabilités, XXII, Lecture Notes in Math. 1321, Springer, Berlin, 1988, pp. 316-347, https://doi.org/10.1007/BFb0084145. · Zbl 0651.47031 |
| [56] | D. Thomson, W. Physick, and R. Maryon, Treatment of interfaces in random walk dispersion models, J. Appl. Meteorol., 36 (1997), pp. 1284-1295. |
| [57] | G. Uffink, A random walk method for the simulation of macrodispersion in a stratified aquifer, in Relation of Groundwater Quantity and Quality, IAHS Publication 146, International Association of Hydrological Sciences, Wallingford, United Kingdom, 1985, pp. 103-114. |
| [58] | J. B. Walsh, A diffusion with discontinuous local time, in Temps locaux, Vol. 52-53, Société Mathématique de France, Paris, 1978, pp. 37-45. |
| [59] | S. Wang, S. Song, and Y. Wang, Skew Ornstein-Uhlenbeck processes and their financial applications, J. Comput. Appl. Math., 273 (2015), pp. 363-382, https://doi.org/10.1016/j.cam.2014.06.023. · Zbl 1304.60046 |
| [60] | E. Zauderer, Partial Differential Equations of Applied Mathematics, 3rd ed., Pure Appl. Math., Wiley, Hoboken, NJ, 2006, https://doi.org/10.1002/9781118033302. · Zbl 1103.35001 |
| [61] | M. Zhang, Calculation of diffusive shock acceleration of charged particles by skew Brownian motion, Astrophys. J., 541 (2000), pp. 428-435. |
| [62] | C. Zheng and G. D. Bennett, Applied Contaminant Transport Modelling, 2nd ed., Wiley-Interscience, New York, 2002. |
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