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

Analyses and Applications of the Peaceman--Rachford and Douglas--Rachford Splitting Schemes

Splitting methods are widely used as temporal discretizations of evolution equations. Such methods usually constitute competitive choices whenever a vector field can be split into a sum of two or more parts that each generates a flow easier to compute or approximate than the flow of the sum. In the research presented in this Licentiate thesis we consider dissipative evolution equations with vector fields given by unbounded operators. Dynamical systems that fit into this framework can for example be found among Hamiltonian systems and parabolic and hyperbolic partial differential equations (PDEs). The goal of the presented research is to perform convergence analyses for the lternating direction implicit (ADI) methods in the setting of dissipative operators. In this context these methods are known to possess excellent stability properties. Additionally, they generate easily computable numerical flows and are ideal choices for studying convergence to stationary solutions, a property related to their favorable local error structure. In this thesis we consider the Peaceman--Rachford and Douglas--Rachford schemes, which were the first ADI methods to be constructed and to this day are the most representative members of the ADI method class. We perform convergence studies for the Peaceman--Rachford and Douglas--Rachford schemes when applied to semilinear, dissipative evolution equations, that is, when the vector fields are given by the sum of a linear and a nonlinear dissipative operator. Optimal convergence orders are proven when the solution is sufficiently regular. With less regularity present we are still able to prove convergence, however of suboptimal order or without order. In contrast to previous convergence order analyses we do not assume Lipschitz continuity of the nonlinear operator. In the context of linear, dissipative evolution equations we consider full space-time discretizations. We assume that the full discretization is given by combining one of the two aforementioned ADI methods with a general, converging spatial discretization method. In this setting we prove optimal, simultaneous space-time convergence orders. Advection-diffusion-reaction models, encountered in physics, chemistry, and biology are important examples of dissipative evolution equations. In this thesis we present such a model describing the growth of axons in nerve cells. The model consists of a parabolic PDE, which has a non-trivial coupling to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. Since additionally the biological model parameters imply a wide range of scales, both in time and space, the application of a numerical method is involved. We make an argument for a discretization consisting of a splitting which is integrated by the Peaceman--Rachford scheme. The choice is motivate by the results of some numerical experiments