References
- A. Abdulle and G.A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys. 231 (2012), pp. 2482–2497.
- R.C. Almeida and J.T. Oden, Solution verification, goal-oriented adaptive methods for stochastic advection–diffusion problems, Comput. Methods Appl. Mech. Eng. 199 (2010), pp. 2472–2486.
- C.N. Angstmann, I.C. Donnelly, B.I. Henry, B.A. Jacobs, T.A.M. Langlands, and J.A. Nichols, From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial, differential equations, J. Comput. Phys. 307 (2016), pp. 508–534.
- V. Asokan, B. Narayanan, and N. Zabaras, Variational multiscale stabilized FEM formulations for transport equations: stochastic advection–diffusion and incompressible stochastic Navier-Stokes equations, J. Comput. Phys. 202 (2005), pp. 94–133.
- Benes̃ M., A. Nekvinda, and M.K. Yadav, Multi-time-step domain decomposition method with non-matching grids for parabolic problems, Appl. Math. Comput. 267 (2015), pp. 571–582.
- A.M. Bruaset and A. Tveito, Numerical Solution of Partial Differential Equations on Parallel Computers, Springer, Berlin, 2005.
- M.D. Chekroun, E. Park, and R. Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differ. Equ. 260 (2016), pp. 2926–972.
- C-M. Chen, F. Liu, V. Anh, and I. Turner, Numerical simulation for the variable-order Galilei invariant advection–diffusion equation with a nonlinear source term, Appl. Math. Comput. 217 (2011), pp. 5729–5742.
- R. Company, E. Ponsoda, J.V. Romero, and M.D. Roselló, A second order numerical method for solving advection–diffusion models, Math. Comput. Model. 50 (2009), pp. 806–811.
- G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, 1992, pp. XVIII+454.
- M. El-Amrani, M. Seaid, and M. Zahri, A stabilized finite element method for stochastic incompressible Navier–Stokes equations, Int. J. Comput. Math. 89(18) (2012), pp. 2576–2602.
- R. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
- T. Giletti, Traveling waves for a reaction–diffusion-advection system with interior or boundary losses, C. R. Acad. Sci. Paris, Ser. I 349 (2011), pp. 535–539.
- W. Gropp and B.F. Smith, Scalable, extensible, and portable numerical libraries, Proceedings of the Scalable Parallel Libraries Conference, Argonne National Laboratory, 1993, pp. 87–93.
- H. Holden, B. Oksendal, J. Uboe, and T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, Probability and its Applications, Birkhäuser, Basel, 1996.
- A. Jentzen and P. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, The Society for Industrial and Applied Mathematics, SIAM, Philadelphia, 2011.
- P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer Verlag, Berlin, 1992.
- N. Mai-Duy and T. Tran-Cong, An efficient domain-decomposition pseudo-spectral method for solving elliptic differential equations, Commun. Numer. Meth. Eng 24 (2008), pp. 795–806.
- H. Manouzi, M. Seaïd, and M. Zahri, Wick-stochastic finite element solution of reaction-diffusion problems, J. Comput. Appl. Math. 203 (2007), pp. 516–532.
- T. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, Springer, Berlin, 2008.
- G.N. Milstein and M.V. Tretyakov, Layer methods for stochastic Navier–Stokes equations using simplest characteristics, J. Comput. Appl. Math. 302 (2016), pp. 1–23.
- K. Mohamed, M. Seaïd, and M. Zahri, A finite volume method for scalar conservation laws with stochastic time–space-dependent flux functions, J. Comput. Appl. Math. 237 (2013), pp. 614–632.
- H. Nishikawa, A first-order system approach for diffusion equation. II: Unification of advection and diffusion, J. Comput. Phys. 229 (2010), pp. 3989–4016.
- A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999.
- A. Quarteroni, A. Veneziani, and P. Zunino, A domain decomposition method for advection–diffusion processes with application to blood solutes, SIAM J. Sci. Comput. 23(6) (2002), pp. 1959–1980.
- A. Rößler, Runge–Kutta methods for Itô stochastic differential equations with scalar noise, BIT 46(1) (2006), pp. 97–110.
- A. Rößler, M. Seaïd, and M. Zahri, Method of lines for stochastic boundary-value problems with additive noise, App. Math. Comput. 199 (2008), pp. 301–314.
- A. Rößler, M. Seaïd, and M. Zahri, Numerical simulation of stochastic replicator models in catalyzed RNA-like polymers, Math. Comput. Simul. 79 (2009), pp. 3577–3586.
- B.L. Rozovskii, Stochastic Evolution Systems, Linear Theory and Application to Nonlinear Filtering, Kluwer Academic Publishers, Dordrecht, 1991.
- W.E. Schiesser, The Numerical Method of Lines. Integration of Partial Differential Equations, Academic Press, San Diego, CA, 1991.
- H.A. Schwartz, Über einen grenzübergang durch alternirendes Verfahren, Ges. Math. Abh. Bd.1 Berlin 1870, s133–143.
- W. Shao and X. Wu, An effective Chebyshev tau meshless domain decomposition method based on the integration-differentiation for solving fourth order equations, Appl. Math. Model. 39 (2015), pp. 2554–2569.
- B.F. Smith and O.B. Widlund, A domain decomposition algorithm using a hierarchical basis, SIAM J. Scient. Statist. Comput. 11(6) (1990), pp. 1212–1220.
- B. Smith, P. Bjorstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Europe-North America, 2004, ISBN 0-521-49589-X.
- J.G. Verwer and J.M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations, Computing 33(3-4) (1984), pp. 297–313.
- A. Zafarullah, Application of the method of lines to parabolic partial differential equations with error estimates, J. Assoc. Comput. Mach. 17(2) (1970), pp. 294–302.
- M. Zahri, Colored-noise-like multiple itô stochastic integrals: algorithms and numerics, J. Numer. Math. Stochast. 7(1) (2015), pp. 48–69.