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Numerical analysis of the stochastic moving boundary problem. (English) Zbl 1267.60078

In [Ill. J. Math. 54, No. 3, 927–962 (2010; Zbl 1264.35285)], the authors proved existence and uniqueness for the following problem \[ du(t,x)=\frac{\partial^2u}{\partial x^2}(t,x)dt+\alpha u(t,x)dt+u(t,x)\circ dB_t, \quad x>\beta(t) \] with \[ \lim_{x\searrow \beta(t)}\frac{\partial u}{\partial x}(t,x)=1,\;u(0,x)=u_{0}(x)\text{ for }x\in \mathbb{R} \] and \[ \{(t,x)\in \mathbb{R}_+\times \mathbb{R} \mid u(t,x)>0\}=\{(t,x)\in \mathbb{R}_+\times \mathbb{R}\mid x>\beta(t)\}. \] In the present paper, a numerical approximation of the solution to this problem is proposed through a spatial shift which leads to a nonlinear stochastic partial differential equation (SPDE) on a fixed domain. A truncation method is used to handle the unboundedness of this domain. Numerical approximations for the truncated nonlinear SPDE are obtained using the explicit finite difference method to discretize space and the Euler-Maruyama scheme to discretize time. Some simulation results are presented in the last section.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R35 Free boundary problems for PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 1264.35285
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References:

[1] Barbu V., Probability Theory Related Fields 124 pp 544– (2002) · Zbl 1101.60040 · doi:10.1007/s00440-002-0232-4
[2] DOI: 10.1214/08-AOP408 · Zbl 1162.76054 · doi:10.1214/08-AOP408
[3] DOI: 10.1002/cpa.20094 · Zbl 1093.35010 · doi:10.1002/cpa.20094
[4] Caffarelli L., A geometric approach to free boundary problems (2005) · Zbl 1083.35001 · doi:10.1090/gsm/068
[5] DOI: 10.2307/2154895 · Zbl 0814.35149 · doi:10.2307/2154895
[6] Crank J., Free and Moving Boundary Problems (1984)
[7] Da Prato , G. , and Röckner , M. 2004 . Invariant measures for a stochastic porous medium equation. InStochastic analysis and related topics in Kyoto.volume 41 of Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, 13–29. · Zbl 1102.76074
[8] Da Prato G., Journal of Evolution Equations 4 pp 249– (2004) · Zbl 1049.60092 · doi:10.1007/s00028-003-0140-9
[9] DOI: 10.1080/03605300500357998 · Zbl 1158.60356 · doi:10.1080/03605300500357998
[10] Elliott C. M., Weak and Variational Methods for Moving Boundary Problems (1982) · Zbl 0476.35080
[11] Gaines , J. G. 1995 . Numerical experiments with S(P)DE’s. InStochastic partial differential equations.Volume 216 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 55–71. · Zbl 0829.60048
[12] Givoli D., Numerical Methods for Problems in Infinite Domains (1992) · Zbl 0788.76001
[13] DOI: 10.1137/S0036144500378302 · Zbl 0979.65007 · doi:10.1137/S0036144500378302
[14] DOI: 10.1016/j.jde.2005.02.006 · Zbl 1099.35187 · doi:10.1016/j.jde.2005.02.006
[15] DOI: 10.1090/S0002-9947-10-04945-7 · Zbl 1197.35290 · doi:10.1090/S0002-9947-10-04945-7
[16] Kim K., Illinois Journal of Mathematics.
[17] Kim K., Journal of Theoretical Probability.
[18] Stewart D.S., Journal de Mechanique Theorique et Applique 4 pp 103– (1985)
[19] Vazquez , J.L. 1996 . The free boundary problem for the heat equation with fixed gradient condition. InFree Boundary Problems, Theory and Applications (Zakopane, 1995).volume 363 ofPitman Research Notes Mathematics Series, Longman, Harlow, 277–302. · Zbl 0867.35120
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