21 research outputs found
Positivity-preserving schemes for some nonlinear stochastic PDEs
We introduce a positivity-preserving numerical scheme for a class of
nonlinear stochastic heat equations driven by a purely time-dependent Brownian
motion. The construction is inspired by a recent preprint by the authors where
one-dimensional equations driven by space-time white noise are considered. The
objective of this paper is to illustrate the properties of the proposed
integrators in a different framework, by numerical experiments and by giving
convergence results
Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting
The efficiency of exact simulation methods for the reaction-diffusion master
equation (RDME) is severely limited by the large number of diffusion events if
the mesh is fine or if diffusion constants are large. Furthermore, inherent
properties of exact kinetic-Monte Carlo simulation methods limit the efficiency
of parallel implementations. Several approximate and hybrid methods have
appeared that enable more efficient simulation of the RDME. A common feature to
most of them is that they rely on splitting the system into its reaction and
diffusion parts and updating them sequentially over a discrete timestep. This
use of operator splitting enables more efficient simulation but it comes at the
price of a temporal discretization error that depends on the size of the
timestep. So far, existing methods have not attempted to estimate or control
this error in a systematic manner. This makes the solvers hard to use for
practitioners since they must guess an appropriate timestep. It also makes the
solvers potentially less efficient than if the timesteps are adapted to control
the error. Here, we derive estimates of the local error and propose a strategy
to adaptively select the timestep when the RDME is simulated via a first order
operator splitting. While the strategy is general and applicable to a wide
range of approximate and hybrid methods, we exemplify it here by extending a
previously published approximate method, the Diffusive Finite-State Projection
(DFSP) method, to incorporate temporal adaptivity
Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System
This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities
Iterative and Non-iterative Splitting approach of a stochastic Burgers' equation
In this paper we present iterative and noniterative splitting methods, which
are used to solve stochastic Burgers' equations. The non-iterative splitting
methods are based on Lie-Trotter and Strang-splitting methods, while the
iterative splitting approaches are based on the exponential integrators for
stochastic differential equations. Based on the nonlinearity of the Burgers'
equation, we have investigated that the iterative schemes are more accurate and
efficient as the non-iterative methods.Comment: 23 pages, 8 figure
Applying splitting methods with complex coefficients to the numerical integration of unitary problems
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr¨odinger equation.
We prove that a particular class of integrators are conjugate to unitary methods
for sufficiently small step sizes when applied to problems defined in the group
SU(2). In the general case, the error in both the energy and the norm of the
numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral
discretization techniques in space