Numerical solution of the Burgers equation with Neumann boundary noise

In this paper we investigate the numerical solution of the one-dimensional Burgers equation with Neumann boundary noise. For the discretization scheme we use the Galerkin approximation in space and the exponential Euler method in time. The impact of the boundary noise on the solution is discussed in several numerical examples. Moreover, we analyze and illustrate some properties of the stochastic term and study the convergence numerically.

[1]  Thorsten Gerber,et al.  Semigroups Of Linear Operators And Applications To Partial Differential Equations , 2016 .

[2]  J. B. Walsh,et al.  On numerical solutions of the stochastic wave equation , 2006 .

[3]  Annie Millet,et al.  Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations , 2007, 0706.1404.

[4]  P. Kloeden,et al.  Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  R. Sowers.  Multidimensional Reaction-Diffusion Equations with White Noise Boundary Perturbations , 1994 .

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

[7]  Jinqiao Duan,et al.  A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions , 2011, Appl. Math. Comput..

[8]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[9]  J. Zabczyk,et al.  Ergodicity for Infinite Dimensional Systems: Invariant measures for stochastic evolution equations , 1996 .

[10]  Nicolai V. Krylov,et al.  An Accelerated Splitting-up Method for Parabolic Equations , 2005, SIAM J. Math. Anal..

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

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

[13]  I. Gyöngy,et al.  On numerical solution of stochastic partial differential equations of elliptic type , 2006 .

[14]  Z. Brzeźniak,et al.  Hyperbolic Equations with Random Boundary Conditions , 2010 .

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

[16]  Jerzy Zabczyk,et al.  Stochastic Partial Differential Equations with Lévy Noise: References , 2007 .

[17]  A. Balakrishnan.  Identification and Stochastic Control of a Class of Distributed Systems with Boundary Noise , 1975 .

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

[19]  Jerzy Zabczyk,et al.  Ergodicity for Infinite Dimensional Systems: Appendices , 1996 .

[20]  A. Debussche,et al.  Numerical simulation of the stochastic Korteweg-de Vries equation , 1999 .

[21]  Predictability of the Burgers dynamics under model uncertainty , 2006, math/0607357.