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The discontinuous Galerkin method for stochastic differential equations driven by additive noises. (English) Zbl 1441.65012

Summary: In this paper, we present a discontinuous Galerkin (DG) finite element method for stochastic ordinary differential equations (SDEs) driven by additive noises. First, we construct a new approximate SDE whose solution converges to the solution of the original SDE in the mean-square sense. The new approximate SDE is obtained from the original SDE by approximating the Wiener process with a piecewise constant random process. The new approximate SDE is shown to have better regularity which facilitates the convergence proof for the proposed scheme. We then apply the DG method for deterministic ordinary differential equations (ODEs) to approximate the solution of the new SDE. Convergence analysis is presented for the numerical solution based on the standard DG method for ODEs. The orders of convergence are proved to be one in the mean-square sense, when \(p\)-degree piecewise polynomials are used. Finally, we present several numerical examples to validate the theoretical results. Unlike the Monte Carlo method, the proposed scheme requires fewer sample paths to reach a desired accuracy level.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), Dover: Dover New York
[2] Baccouch, M., Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations, Appl. Numer. Math., 106, 129-153 (2016) · Zbl 1382.65220
[3] Baccouch, M.; Johnson, B., A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations, J. Comput. Appl. Math., 308, 138-165 (2016) · Zbl 1345.60066
[4] Burrage, K.; Burrage, P., High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 22, 81-101 (1996) · Zbl 0868.65101
[5] Cao, Y.; Yin, L., Spectral Galerkin method for stochastic wave equations driven by space-time white noise, Commun. Pure Appl. Anal., 6, 3, 607-617 (2007) · Zbl 1138.65005
[6] Cheng, Y.; Shu, C.-W., Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47, 4044-4072 (2010) · Zbl 1208.65137
[7] Chow, P.; Jiang, J.; Menaldi, J.-L., Pathwise convergence of approximate solutions to Zakai’s equation in a bounded domain, (DaPrato, G.; Tubaro, L., Proceedings of the Third Conference on Stochastic Partial Differential Equations and Applications. Proceedings of the Third Conference on Stochastic Partial Differential Equations and Applications, Trento, Italy, 1990. Proceedings of the Third Conference on Stochastic Partial Differential Equations and Applications. Proceedings of the Third Conference on Stochastic Partial Differential Equations and Applications, Trento, Italy, 1990, Pitman Research Notes in Mathematics Series, vol. 268 (1992), Longman: Longman London) · Zbl 0789.60040
[8] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland Pub. Co.: North-Holland Pub. Co. Amsterdam-New York-Oxford · Zbl 0383.65058
[9] Dalang, R., A Mini-Course on Stochastic Partial Differential Equations (2009), Springer: Springer Berlin
[10] Gyöngy, I., Approximations of stochastic partial differential equations, (Stochastic Partial Differential Equations and Applications. Stochastic Partial Differential Equations and Applications, Trento, 2002. Stochastic Partial Differential Equations and Applications. Stochastic Partial Differential Equations and Applications, Trento, 2002, Lecture Notes in Pure and Applied Mathematics, vol. 227 (2002), Dekker: Dekker New York), 287-307 · Zbl 1004.60073
[11] Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546 (2001) · Zbl 0979.65007
[12] Higham, D. J.; Mao, X.; Stuart, A. M., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40, 1041-1063 (2002) · Zbl 1026.65003
[13] Ito, K., Approximation of the Zakai equation for nonlinear filtering, SIAM J. Control Optim., 34, 2, 620-634 (1996) · Zbl 0847.93061
[14] Kloeden, P.; Platen, E., Numerical Solution of Stochastic Differential Equations, Stochastic Modelling and Applied Probability (2010), Springer Berlin Heidelberg · Zbl 1216.60052
[15] Kröker, I.; Rohde, C., Finite volume schemes for hyperbolic balance laws with multiplicative noise, Appl. Numer. Math., 62, 4, 441-456 (2012) · Zbl 1241.65013
[16] Liu, D., Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise, Commun. Math. Sci., 1, 2, 361-375 (2003) · Zbl 1086.60037
[17] Mao, X., Stochastic Differential Equations and Applications (2008), Horwood Pub: Horwood Pub Chichester
[18] Matthies, H. G.; Keese, A., Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis. Special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis, Comput. Methods Appl. Mech. Eng., 194, 12-16, 1295-1331 (2005) · Zbl 1088.65002
[19] Meng, X.; Shu, C.-W.; Zhang, Q.; Wu, B., Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM J. Numer. Anal., 50, 5, 2336-2356 (2012) · Zbl 1267.65115
[20] Milstein, G. N.; Tretyakov, M. V., Stochastic Numerics for Mathematical Physics (2004), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg · Zbl 1085.60004
[21] ming Yao, R.; jun Bo, L., Discontinuous Galerkin method for elliptic stochastic partial differential equations on two and three dimensional spaces, Sci. China Ser. A: Math., 50, 11, 1661-1672 (2007) · Zbl 1132.60051
[22] Mohamed, K.; Seaid, M.; Zahri, M., A finite volume method for scalar conservation laws with stochastic time-space dependent flux function, J. Comput. Appl. Math., 237, 1, 614-632 (2013) · Zbl 1252.65152
[23] Oksendal, B., Stochastic Differential Equations: An Introduction with Applications (2010), Springer
[24] Papanicolaou, G. C., Wave propagation in a one-dimensional random medium, SIAM J. Appl. Math., 21, 1, 13-18 (1971) · Zbl 0205.56004
[25] Peszat, S., The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ., 2, 3, 383-394 (2002) · Zbl 1375.60109
[26] Platen, E., An introduction to numerical methods for stochastic differential equations, Acta Numer., 8, 197-246 (1999) · Zbl 0942.65004
[27] Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation (1991), Los Alamos Scientific Laboratory: Los Alamos Scientific Laboratory Los Alamos, Tech. Rep. lA-UR-73-479
[28] Roth, C., A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations, Stoch. Anal. Appl., 24, 1, 221-240 (2006) · Zbl 1086.60038
[29] Sun, M.; Glowinski, R., Pathwise approximation and simulation for the Zakai filtering equation through operator splitting, Calcolo, 30, 3, 219-239 (1993) · Zbl 0819.65149
[30] Watanabe, S.; Ikeda, N., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland Mathematical Library, Elsevier Science · Zbl 0495.60005
[31] Wolfersdorf, L. V., Wave propagation in solids and fluids, J. Appl. Math. Mech./Z. Angew. Math. Mech., 70, 10 (1990), 473-473
[32] Wong, E.; Zakai, M., On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 1560-1564 (1965) · Zbl 0138.11201
[33] Wong, E.; Zakai, M., On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3, 2, 213-229 (1965) · Zbl 0131.16401
[34] Yan, Y., Semi-discrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT Numer. Math., 44, 4, 829-847 (2004) · Zbl 1080.65006
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