Numerical Continuation and SPDE Stability for the 2D CubicQuintic AllenCahn Equation
Abstract
We study the AllenCahn equation with a cubicquintic nonlinear term and a stochastic $Q$traceclass stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand, how numerical continuation methods can be carried over to the SPDE setting. First, we compute the deterministic bifurcation diagram for the PDE, i.e. without stochastic forcing. In this case, two locally asymptotically stable steady state solution branches exist upon variation of the linear damping term. Then we consider the Lyapunov operator equation for the locally linearized system around steady states for the SPDE. We discretize the full SPDE using a combination of finitedifferences and spectral noise approximation obtaining a finitedimensional system of stochastic ordinary differential equations (SODEs). The large system of SODEs is used to approximate the Lyapunov operator equation via covariance matrices. The covariance matrices are numerically continued along the two bifurcation branches. We show that we can quantify the stochastic fluctuations along the branches. We also demonstrate scaling laws near branch and fold bifurcation points. Furthermore, we perform computational tests to show that, even with a suboptimal computational setup, we can quantify the subexponentialtimescale fluctuations near the deterministic steady states upon stochastic forcing on a standard desktop computer setup. Hence, the proposed method for numerical continuation of SPDEs has the potential to allow for rapid parametric uncertainty quantification of spatiotemporal stochastic systems.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.4000
 Bibcode:
 2014arXiv1408.4000K
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Physics  Computational Physics
 EPrint:
 revised version, 30 pages, 11 figures [movie not included due to arXiv size limitations]