Global random walk on grid algorithm for solving Navier–Stokes and Burgers equations

References

[1] H. Amann, Eine Monte-Carlo-Methode mit Informationsspeicherung zur Lösung von elliptischen Randwertproblemen, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 8 (1967), 117–130. 10.1007/BF00536914Search in Google Scholar

[2] G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford, 1976. Search in Google Scholar

[3] A. F. Cheshkova, Global estimate of the solution of the Dirichlet problem for the Helmholtz 𝑛-dimensional equation by the Monte Carlo method, Russian J. Numer. Anal. Math. Modelling 10 (1995), no. 6, 495–510. 10.1515/rnam.1995.10.6.495Search in Google Scholar

[4] A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. 10.1017/S0022112073002016Search in Google Scholar

[5] N. R. Clarke and O. R. Tutty, Construction and validation of a discrete vortex method for the two-dimensional incompressible Navier–Stokes equations, Comput. & Fluids 23 (1994), no. 6, 751–783. 10.1016/0045-7930(94)90065-5Search in Google Scholar

[6] S. M. Ermakov, V. V. Nekrutkin and A. S. Sipin, Random Processes for Classical Equations of Mathematical Physics, Kluwer Academic, Dordrecht, 1989. 10.1007/978-94-009-2243-3Search in Google Scholar

[7] S. Karl, First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations, Appl. Math. Lett. 88 (2019), 141–148. 10.1016/j.aml.2018.08.018Search in Google Scholar

[8] A. Kireeva, K. K. Sabelfeld and S. Kireev, Parallel simulation of drift-diffusion-recombination by cellular automata and global random walk algorithm, J. Supercomput. 77 (2021), 6889–6903. 10.1007/s11227-020-03529-ySearch in Google Scholar

[9] J. Lequeurre and A. Munnier, Vorticity and stream function formulations for the 2D Navier–Stokes equations in a bounded domain, J. Math. Fluid Mech. 22 (2020), no. 2, Paper No. 15. 10.1007/s00021-019-0479-5Search in Google Scholar

[10] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University, Cambridge, 1995. 10.1017/CBO9780511814075Search in Google Scholar

[11] K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer, Berlin, 1991. 10.1007/978-3-642-75977-2Search in Google Scholar

[12] K. K. Sabelfeld, Stochastic algorithms in linear algebra—beyond the Markov chains and von Neumann–Ulam scheme, Numerical Methods and Applications, Lecture Notes in Comput. Sci. 6046, Springer, Heidelberg (2011), 14–28. 10.1007/978-3-642-18466-6_2Search in Google Scholar

[13] K. K. Sabelfeld, Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems, Monte Carlo Methods Appl. 23 (2017), no. 3, 189–212. 10.1515/mcma-2017-0113Search in Google Scholar

[14] K. K. Sabelfeld, A global random walk on spheres algorithm for transient heat equation and some extensions, Monte Carlo Methods Appl. 25 (2019), no. 1, 85–96. 10.1515/mcma-2019-2032Search in Google Scholar

[15] K. K. Sabelfeld, Stochastic simulation algorithms for solving narrow escape diffusion problems by introducing a drift to the target, J. Comput. Phys. 410 (2020), Article ID 109406. 10.1016/j.jcp.2020.109406Search in Google Scholar

[16] K. K. Sabelfeld, V. Kaganer, C. Pfueller and O. Brandt, Dislocation contrast in cathodoluminescence and electron-beam induced current maps on GaN(0001), J. Phys. D 50 (2017), no. 40, Article ID 405101. Search in Google Scholar

[17] K. K. Sabelfeld and A. Kireeva, A new global random walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of points, Appl. Math. Lett. 107 (2020), Article ID 106466. 10.1016/j.aml.2020.106466Search in Google Scholar

[18] K. K. Sabelfeld and A. Kireeva, Parallel implementation of stochastic simulation algorithm for nonlinear systems of electron-hole transport equations in a semiconductor, Parallel Computational Technologies 2020, Commun. Comput. Inf. Sci. 1263, Springer, Cham (2020), 251–266. 10.1007/978-3-030-55326-5_18Search in Google Scholar

[19] K. K. Sabelfeld and N. Loshchina, Stochastic iterative projection methods for large linear systems, Monte Carlo Methods Appl. 16 (2010), no. 3–4, 343–359. 10.1515/mcma.2010.020Search in Google Scholar

[20] K. K. Sabelfeld and N. S. Mozartova, Sparsified randomization algorithms for low rank approximations and applications to integral equations and inhomogeneous random field simulation, Math. Comput. Simulation 82 (2011), no. 2, 295–317. 10.1016/j.matcom.2011.08.002Search in Google Scholar

[21] K. K. Sabelfeld and D. Smirnov, A global random walk on grid algorithm for second order elliptic equations, Monte Carlo Methods Appl. 27 (2021), no. 3, 211–225. 10.1515/mcma-2021-2092Search in Google Scholar

[22] K. K. Sabelfeld, D. Smirnov, I. Dimov and V. Todorov, A global random walk on grid algorithm for second order elliptic equations, Monte Carlo Methods Appl. 27 (2021), no. 4, 325–339. 10.1515/mcma-2021-2097Search in Google Scholar

[23] H. S. Shukla, M. Tamsir, V. K. Srivastava and and J. Kumar, Numerical solution of two dimensional coupled viscous Burgers’ equation using modified cubic B-spline differential quadrature method, AIP Adv. 4 (2014), 10.1063/1.4902507. 10.1063/1.4902507Search in Google Scholar

[24] V. K. Srivastava and U. Bhardwaj, Numerical solutions to Burgers’ equations, Thesis, 2011. Search in Google Scholar

[25] T. Strohmer and R. Vershynin, A randomized Kaczmarz algorithm with exponential convergence, J. Fourier Anal. Appl. 15 (2009), no. 2, 262–278. 10.1007/s00041-008-9030-4Search in Google Scholar

[26] H. Zhu, H. Shu and M. Ding, Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method, Comput. Math. Appl. 60 (2010), no. 3, 840–848. 10.1016/j.camwa.2010.05.031Search in Google Scholar