A full space-time convergence order analysis of operator splittings for linear dissipative evolution equations

The Douglas--Rachford and Peaceman--Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments

Efficient simulations of tubulin-driven axonal growth

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman--Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g.\ its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.Comment: Authors' accepted version, (post refereeing). The final publication (in Journal of Computational Neuroscience) is available at Springer via http://dx.doi.org/10.1007/s10827-016-0604-

Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which also contains numerical experiments. Version 3 and 4: Only comments added. Version 2, page 2: Clarified statement on stability issues for ADI schemes with more than two operator

Spatial and Physical Splittings of Semilinear Parabolic Problems

Splitting methods are widely used temporal approximation schemes for parabolic partial differential equations (PDEs). These schemes may be very efficient when a problem can be naturally decomposed into multiple parts. In this thesis, splitting methods are analysed when applied to spatial splittings (partitions of the computational domain) and physical splittings (separations of physical processes) of semilinear parabolic problems. The thesis is organized into three major themes: optimal convergence order analysis, spatial splittings and a physical splitting application.In view of the first theme, temporal semi-discretizations based on splitting methods are considered. An analysis is performed which yields convergence without order under weak regularity assumptions on the solution, and convergence orders ranging up to classical for progressively more regular solutions. The analysis is performed in the framework of maximal dissipative operators, which includes a large number of parabolic problems. The temporal results are also combined with convergence studies of spatial discretizations to prove simultaneous space–time convergence orders for full discretizations.For the second theme, two spatial splitting formulations are considered. For dimension splittings each part of the formulation represents the evolution in one spatial dimension only. Thereby, multidimensional problems can be reduced to families of one-dimensional problems. For domain decomposition splittings each part represents a problem on only a smaller subdomain of the full domain of the PDE. The results of the first theme are applied to prove optimal convergence orders for splitting schemes used in conjunction with these two splitting formulations. The last theme concerns the evaluation of a physical splitting procedure in an interdisciplinary application. A model for axonal growth out of nerve cells is considered. This model features several challenges to be addressed by a successful numerical method. It consists of a linear PDE coupled to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. The biological model parameters imply a wide range of scales, both in time and space. Based on a physical splitting, a tailored scheme for this model is constructed. Its robustness and efficiency are then verified by numerical experiments

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

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

Approximate factorization for time-dependent partial differential equations

AbstractThe first application of approximate factorization in the numerical solution of time-dependent partial differential equations (PDEs) can be traced back to the celebrated papers of Peaceman and Rachford and of Douglas of 1955. For linear problems, the Peaceman–Rachford–Douglas method can be derived from the Crank–Nicolson method by the approximate factorization of the system matrix in the linear system to be solved. This factorization is based on a splitting of the system matrix. In the numerical solution of time-dependent PDEs we often encounter linear systems whose system matrix has a complicated structure, but can be split into a sum of matrices with a simple structure. In such cases, it is attractive to replace the system matrix by an approximate factorization based on this splitting. This contribution surveys various possibilities for applying approximate factorization to PDEs and presents a number of new stability results for the resulting integration methods