Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition

Abstract In this paper we obtain a general statement concerning pathwise convergence of the full discretization of certain stochastic partial differential equations (SPDEs) with non-globally Lipschitz continuous drift coefficients. We focus on non-diagonal colored noise instead of the usual space–time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen, Kloeden and Winkel we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate.

[1]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[2]  Tony Shardlow,et al.  Postprocessing for Stochastic Parabolic Partial Differential Equations , 2007, SIAM J. Numer. Anal..

[3]  E. Hausenblas.  Numerical analysis of semilinear stochastic evolution equations in Banach spaces , 2002 .

[4]  Arnulf Jentzen,et al.  Galerkin Approximations for the Stochastic Burgers Equation , 2013, SIAM J. Numer. Anal..

[5]  Hyek Yoo,et al.  Semi-discretization of stochastic partial differential equations on R1 by a finite-difference method , 2000, Math. Comput..

[6]  S. Cox,et al.  Pathwise Holder convergence of the implicit Euler scheme for semi-linear SPDEs with multiplicative noise , 2012, 1201.4465.

[7]  Arnulf Jentzen,et al.  Efficient simulation of nonlinear parabolic SPDEs with additive noise , 2011, 1210.8320.

[8]  Di Liu.  Convergence of the Spectral Method for Stochastic Ginzburg-Landau Equation Driven by Space-Time White Noise , 2003 .

[9]  P. Kloeden,et al.  LINEAR-IMPLICIT STRONG SCHEMES FOR ITO-GALKERIN APPROXIMATIONS OF STOCHASTIC PDES , 2001 .

[10]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[11]  D. Blömker.  Nonhomogeneous Noise and Q-Wiener Processes on Bounded Domains , 2005 .

[12]  I. Gyöngy.  Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II , 1999 .

[13]  S. Hosseini,et al.  Full discretization of the stochastic Burgers equation with correlated noise , 2013 .

[14]  D. Nualart,et al.  On the Stochastic Burgers’ Equation in the Real Line , 1999 .

[15]  Dariusz Gatarek,et al.  Stochastic burgers equation with correlated noise , 1995 .

[16]  Jan van Neerven,et al.  Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise , 2013, Numerische Mathematik.

[17]  Arnulf Jentzen.  Pathwise Numerical Approximations of SPDEs with Additive Noise under Non-global Lipschitz Coefficients , 2009 .

[18]  T. Shardlow.  Numerical methods for stochastic parabolic PDEs , 1999 .