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Arif ULLAH, Majid KHAN, M KAMRAN, R KHAN, Zhengmao SHENG (盛正卯). Monte-Carlo simulation of a stochastic differential equation[J]. Plasma Science and Technology, 2017, 19(12): 125001. DOI: 10.1088/2058-6272/aa8f3f
Citation: Arif ULLAH, Majid KHAN, M KAMRAN, R KHAN, Zhengmao SHENG (盛正卯). Monte-Carlo simulation of a stochastic differential equation[J]. Plasma Science and Technology, 2017, 19(12): 125001. DOI: 10.1088/2058-6272/aa8f3f

Monte-Carlo simulation of a stochastic differential equation

Funds: This publication is based on the research that has been supported in part by the Higher Education Commission of Pakistan under PPCR program. As well as this, work was supported by the National Magnetic Confinement Fusion Program under Grant No. 2013GB104004 and Fundamental Research Fund for Chinese Central Universities.
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  • Received Date: May 23, 2017
  • For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient, Monte Carlo (MC) techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem. Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.
  • [1]
    ?ksendal B 2003 Stochastic Differential Equations: An Introduction with Applications (Berlin: Springer)
    [2]
    Madera A G 1993 Appl. Math. Modelling 17 664
    [3]
    Mi?kisz J, Gomu?kiewicz J and Mi?kisz S 2014 Math. Appl. 42 39
    [4]
    Higham D J 2001 SIAM Rev. 43 525
    [5]
    Mao Z J et al 2012 Int. J. Eng. Res. Appl. 2 2463
    [6]
    Jacobs K 2010 Stochastic Processes for Physicists: Understanding Noisy Systems (Cambridge: Cambridge University Press)
    [7]
    Einstein A 1905 Ann. Phys. 17 549
    [8]
    von Smoluchowski M 1906 Ann. Phys. 326 756
    [9]
    It? K 1950 Nagoya Math. J. 1 35
    [10]
    It? K 1951 Nagoya Math. J. 3 55
    [11]
    Brissaud A and Frisch U 1974 J. Math. Phys. 15 524
    [12]
    Dostal L and Kreuzer E J 2016 Proc. IUTAM 19 178
    [13]
    Ladde G S and Sambandham M 2003 Stochastic Versus Deterministic Systems of Differential Equations (New York: CRC Press)
    [14]
    Fisch N J 1987 Rev. Mod. Phys. 59 175
    [15]
    Bizarro J P S, Belo J H and Figueiredo A C 1997 Phys. Plasmas 4 2027
    [16]
    Belo J H and Bizarro J P S 2002 IEEE Trans. Plasma Sci. 30 70
    [17]
    Donoso J M and Salgado J J 2006 J. Phys. A: Math. Gen. 39 12587
    [18]
    Kloeden P E and Platen E 1999 Numerical Solution of Stochastic Differential Equations (Berlin: Springer)
    [19]
    Borovkov K 2014 Elements of Stochastic Modelling 2nd edn (Singapore: World Scienti?c)
    [20]
    Mamontov Y and Willander M 2001 High-Dimensional Nonlinear Diffusion Stochastic Processes: Modelling for Engineering Applications (Singapore: World Scienti?c)
    [21]
    Hoffman J D and Frankel S 2001 Numerical Methods for Engineers and Scientists (Boca Raton, FL: CRC Press)
    [22]
    Spigler R 1987 Math. Comput. Simul. 29 243
    [23]
    Homeier D et al 2008 Monte-Carlo-Simulations of Stochastic Differential Equations at the Example of the Forced Burgers’ Equation (Singapore: World Scienti?c) p 346
    [24]
    Graham C and Talay D 2013 Stochastic Simulation and Monte Carlo Methods (Berlin: Springer)
    [25]
    Komori Y, Saito Y and Mitsui T 1994 Comput. Math. Appl. 28 269
    [26]
    Barlett V R, Hoyuelos M and MáRtin H O 2013 J. Comput. Phys. 239 51
    [27]
    Spitzer L 1962 Physics of Fully Ionized Gases (New York: Wiley)
    [28]
    Tautz R C 2013 Astron. Astrophys. 558 A148
    [29]
    Peeters A G and Strintzi D 2008 Ann. Phys. 17 142
    [30]
    Zharovsky E et al 2012 Appl. Numer. Math. 62 1554
    [31]
    Nitzan A 2006 Chemical Dynamics in Condensed Phases (Oxford: Oxford University Press)
    [32]
    Mukhtar Q, Hellsten T and Johnson T 2010 IEEE Trans. Plasma Sci. 38 2185
    [33]
    Kloeden P E and Neuenkirch A 2007 J. Comput. Math. 10 235
    [34]
    Kloeden P E and Platen E 1992 Numerical Solution of Stochastic Differential Equations (Berlin: Springer)
    [35]
    Mil’stein G N 1974 Theory Probab. Appl. 19 557
    [36]
    van den Doel K, Ascher U M and Haber E 2013 The lost honor of l2-based regularization ed M Cullen et al Large Scale Inverse Problems (Berlin: De Gruyter) pp 181–203
    [37]
    Mukhtar Q, Hellsten T and Johnson T 2013 Plasma Phys. Control. Fusion 55 095011
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