Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils
Applied and Computational Mathematics
Volume 2, Issue 4, August 2013, Pages: 100-108
Received: May 30, 2013; Published: Aug. 10, 2013
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Taras A. Dauzhenka, R&D Dept., Simmakers Ltd., 1A-307 V. Kharuzhei str., 220005 Minsk, Belarus
Igor A. Gishkeluk, R&D Dept., Simmakers Ltd., 1A-307 V. Kharuzhei str., 220005 Minsk, Belarus
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Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.
Splitting Algorithms, Alternating Directions Scheme, Stefan Problem, Nonlinear Heat Equation, Consistent Approximation
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Taras A. Dauzhenka, Igor A. Gishkeluk, Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils, Applied and Computational Mathematics. Vol. 2, No. 4, 2013, pp. 100-108. doi: 10.11648/j.acm.20130204.11
N. N. Yanenko, The method of fractional steps: The solution of problems of mathematical physics in several variables, 1st ed. Springer- Verlag, (1971).
Ye.G. D'yakonov "On the application of disintegrating difference operators", USSR Computational Mathematics and Mathematical Physics, vol. 3, no.2, (1963), pp. 511 – 515.
A.A. Samarskii "On the convergence of the fractional step method for heat conductivity equation", USSR Computational Mathematics and Mathematical Physics, vol. 2, no.6, (1963). Pp. 1347 – 1354.
A.A. Samarskii "Local one dimensional difference schemes on non-uniform nets". USSR Computational Mathematics and Mathematical Physics, vol. 3, no. 3, (1963), pp. 572 – 619.
G. I. Marchuk "On the theory of the splitting-up method". Proceedings of the 2nd Symposium on Numerical Solution of Partial Differential Equations (1970), SVNPADE, pp. 469 – 500.
J. Douglas, "Alternating direction methods for three space variables," Numerische Mathematik, vol. 4, no. 1, pp. 41–63, Dec. 1962.
J. Douglas and H. H. Rachford, "On the numerical solution of heat conduction problems in two and three space variables," Transaction of the American Mathematical Society, vol. 82, pp. 421–489, 1956.
J. Douglas, R. B. Kellogg, and R. S. Varga, "Alternating direction iteration methods for n space variables," Mathematics of Computation, vol. 17, no. 83, pp. 279–282, 1963.
D. W. Peaceman and H. H. Rachford, "The numerical solution of parabolic and elliptic differential equations," J. Soc. Ind. Appl. Math., vol. 3, pp. 28–41, 1955.
H. B. de Vries, "A comparative study of ADI splitting methods for parabolic equation in two space dimensions," J. Comp. Appl. Math., vol. 10, pp. 179–193, 1984.
G. I. Marchuk, Splitting and Alternating Direction Methods, ser. Finite Difference Methods. North Holland: Elsevier Science Publishers B.V., 1990, vol. 1, pp. 199–462.
S. Ogurtsov, G. Pan, and R. Diaz, "Examination, clarification, and simplification of stability and dispersion analysis for adi-fdtd and cnss-fdtd schemes," Antennas and Propagation, IEEE Transactions on, vol. 55, no. 12, dec. 2007.
K. J. in ’t Hout and B. D. Welfert, "Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms," Applied Numerical Mathematics, vol. 59, no. 3-4, pp. 677–692, Mar. 2009.
J. Qin, "The new alternating direction implicit difference methods for solving three-dimensional parabolic equations," Appl. Math. Model., vol. 34, no. 4, pp. 890–897, 2010.
Yu. N. Skiba, D. M. Filatov "Splitting-based schemes for numerical solution of nonlinear diffusion equations on a sphere" Applied Mathematics and Computation, Vol. 219, no. 16, (2013), pp. 8467 – 8485.
Yu. N. Skiba, D.M. Filatov "Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere". In: Simulation and Modeling Methodologies, Technologies and Applications. Series: Advances in Intelligent Systems and Computing, Vol. 197, Eds.: Pina N. et al., Springer-Verlag Berlin Heidelberg, (2013), pp. 57-70.
P.L. Lions, B. Mercier "Splitting algorithms for the sum of two nonlinear operators", SIAM J. Numer. Anal., Vol. 16, no. 6, (1979), pp. 964 – 979.
J. M. Borwein, B. Sims "The Douglas – Rachford algorithm in the absence of convexity", In Fixed-Point Algorithms for Inverse Problems in Science and Engineering (2011), H.H. Bauschke et al. (eds.), Springer Optimization and Its Applications 49, pp. 93-109.
J. Eckstein "Splitting methods for monotone operators with applications to parallel optimization", June 1989. Report LIDS-TH-1877, Massachusetts Institute of Technology, Cambridge, MA 02139.
J. Geiser Iterative splitting methods for differential equations. Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series, edited by Magoules and Lai, 2011.
H. Holden, K.H. Karlsen, K.-A. Lie, N.H. Risebro Splitting Methods for Partial Differential Equations with Rough Solutions. EMS Series of Lectures in Mathematics (2010).
A.N. Tikhonov, A.A. Samarskii "Homogeneous difference schemes" USSR Computational Mathematics and Mathematical Physics, vol. 1, no. 1, (1962), pp. 5 - 67.
D.W. Peaceman Fundamentals of numerical reservoir simulation. Elsevier SP (1977).
J. Eckstein, D.P. Bertsekas "On the Douglas – Rachford splitting method and the proximal point algorithm for maximal monotone operators" Mathematical Programming 55 (1992), pp. 293 – 318.
G. Birkhoff, R.S. Varga "Implicit alternating direction methods", Trans. Amer. Math. Soc., vol. 92, (1959), pp. 13 - 24.
J.E. Dendy "Alternating direction methods for nonlinear time-dependent problems" SIAM J. Numer. Anal., vol. 14, no. 2, (1977), pp. 313 - 326.
E. Hansen, A. Ostermann "Dimension splitting for quasilinear parabolic equations", IMA J. Numer. Anal., vol. 30, no. 3, (2010), pp. 857 - 869.
M. Schatzman "Stability of the Peaceman - Rachford approximation", J. Functional Analysis., vol. 162, no. 1, (1999), pp. 219 - 255.
B. Bialecki, R. Fernandes "An alternating-direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons". SIAM J. Scientific Computing, vol. 28, no. 3, (2006), pp. 1054 - 1077.
R.B. Kellog "Nonlinear alternating direction algorithm" Math. Comp. vol. 23, (1969), pp. 23 – 28.
B. He and X. Yuan "On the O(1/n) convergence rate of the Douglas-Rachford alternative direction method". SIAM J. Numerical Analysis, Vol. 50, no. 2, (2012), pp. 700-709.
B.F. Svaiter "On weak convergence of the Douglas - Rachford method" SIAM J. Control Optim., Vol. 49, no. 1, (2011), pp. 280 – 287.
J. Eckstein, B.F. Svaiter "General projective splitting methods for sums of maximal monotone operators" SIAM J. Control Optim., Vol. 48, no. 2, (2009), pp. 787 – 811.
H. Brezis, A. Pazy "Convergence and approximation of semi-groups of nonlinear operators in Banach spaces", J. Funct. Anal., vol. 9, (1972), pp. 63 – 74.
H. Brezis Operateurs maximaux monotones et semigroupes de contraction dans les espaces de Hilbert, North-Holland Mathematics Studies 5, ed. Leopoldo Nachbin, Amsterdam, 1973.
E. Hille, R.S. Phillips "Functional analysis and semi-groups", AMS Colloquium Publications, vol. 31, 1996.
E. Zeidler Nonlinear functional analysis and its applications: II/B Nonlinear monotone operators, Springer-Verlag, New-York, 1990.
G.J. Minty "Monotone (nonlinear) operators in Hilbert space", Duke Math. J., vol. 29, no.3, (1962), pp. 341 – 346.
O.A. Ladyzhenskaja, V.A. Solonnikov, N.N. Ural’ceva Linear and quasi-linear equations of parabolic type, ser. Translations of Mathematical Monographs. Providence, RI: AMS, 1968, vol. 23.
S. Kamenomostskaya "On the Stefan problem". Mat. Sb., N. Ser., vol. 53(95), no. 4, (1961), pp. 489 – 514.
M. Lees "Alternating direction and semi-explicit difference methods for parabolic partial differential equations", Numerische Mathematik, vol. 3, no. 1, (1961), pp. 398 – 412.
T.A. Dauzhenka, I.A. Gishkeluk "Quasilinear heat equation in three dimensions and Stefan problem in permafrost soils in the frame of alternating directions finite difference scheme" Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2013, U.K., 3 – 5 July, 2013, London, pp. 1 - 6.
A.A. Samarskii, P.N. Vabishchevich, Mathematical Modelling, Vol. 1: Computational Heat Transfer. Wiley, 1996.
V.I. Vasilyev, V.V. Popov "Numerical solution of the soil freezing problem", Math. Models Comput. Simulations, vol. 1, no. 4, (2009), pp. 419 – 427.
A.A. Samarskii, B.D. Moiseyenko "An economic continuous calculation scheme for the Stefan multidimensional problem", USSR Computational Mathematics and Mathematical Physics, vol. 5, no. 5, (1965), pp. 43 – 58.
A.M. Maksimov, G.G. Tsypkin "A mathematical model for the freezing of water-saturated porous medium", USSR Computational Mathematics and Mathematical Physics, vol. 26, no. 6, (1986), pp. 91 – 95.
L. Bronfenbrener, E. Korin, "Two-phase zone formation conditions under freezing of porous media", J. Crystal Growth, vol. 198-199, (1999), pp. 89 – 95.
H.R. Thomas, P. Cleall, Y.C. Li, C. Harris, M.Kern-Leutschg, "Modelling of cryogenic processes in permafrost and seasonally frozen soils", Geotechnique, vol. 59, no. 3, (2009), pp. 173 – 184.
L. Bronfenbrener, R. Bronfenbrener, "Frost heave and phase front instability in freezing soils", Cold Regions Science and Technology, vol. 64, no. 1, (2010), pp. 19 – 38.
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