Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

Numerical solutions of heterogeneous Helmholtz problems present various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work, we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications

Segmentation and Scene Content in Moving Images

The problem of scene content in moving images was brought by Aralia. The goal in this study group was to consider two problems. The first was image segmentation and the second is the context of the scene. These problems were explored in different areas, namely the Bayesian approach to image segmentation, shadow detection, shape recognition and background separation

Scalable two-phase flow solvers

Two-phase flows arise in many areas of application such as in the study of coastal and hydraulic processes. Often the fluids involved can be modelled by incompressible phases which have disparate physical properties such as density and viscosity. We utilise a two-phase flow model of immiscible Newtonian fluids. A key part of this model is a set of variable coefficient Navier–Stokes equations. This thesis focuses on the numerical solution of the linear systems which arise after linearisation of these equations. Solving these systems often dominates the computation time when running simulations. One of the challenges in solving the model is that the density and viscosity coefficients are discontinuous and can have large jumps between the two phases. In this work we consider preconditioned iterative Krylov methods to solve the large and sparse linear systems and pay particular attention to incorporating the highly varying coefficients into the block preconditioners that we propose. We will see that such considerations can be essential in order to obtain good performance. An important issue is the scalability of the solution methodology. Here, we will study how the convergence of the iterative solver depends on a grid parameter which controls the refinement of the computational mesh. We will see that the novel preconditioners we propose can lead to convergence which is effectively independent of the grid parameter. We also investigate dependence on other model parameters such as the Reynolds number as well as the density and viscosity ratios between the two fluids. Another topic we examine is the use of a multipreconditioned iterative method allowing more than one preconditioner to be used simultaneously. Our results using this approach show some promising features. Finally, we consider an implementation within a more realistic model used in practice for simulating complex air–water flows. In particular, we will provide results for a problem modelling the breaking of a dam.</p

Scalable two-phase flow solvers

Two-phase flows arise in many areas of application such as in the study of coastal and hydraulic processes. Often the fluids involved can be modelled by incompressible phases which have disparate physical properties such as density and viscosity. We utilise a two-phase flow model of immiscible Newtonian fluids. A key part of this model is a set of variable coefficient NavierâStokes equations. This thesis focuses on the numerical solution of the linear systems which arise after linearisation of these equations. Solving these systems often dominates the computation time when running simulations. One of the challenges in solving the model is that the density and viscosity coefficients are discontinuous and can have large jumps between the two phases. In this work we consider preconditioned iterative Krylov methods to solve the large and sparse linear systems and pay particular attention to incorporating the highly varying coefficients into the block preconditioners that we propose. We will see that such considerations can be essential in order to obtain good performance. An important issue is the scalability of the solution methodology. Here, we will study how the convergence of the iterative solver depends on a grid parameter which controls the refinement of the computational mesh. We will see that the novel preconditioners we propose can lead to convergence which is effectively independent of the grid parameter. We also investigate dependence on other model parameters such as the Reynolds number as well as the density and viscosity ratios between the two fluids. Another topic we examine is the use of a multipreconditioned iterative method allowing more than one preconditioner to be used simultaneously. Our results using this approach show some promising features. Finally, we consider an implementation within a more realistic model used in practice for simulating complex airâwater flows. In particular, we will provide results for a problem modelling the breaking of a dam.</p

On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices. ETNA - Electronic Transactions on Numerical Analysis

In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm being applied to the equations with absorption, then, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that are typically encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments

A comparison of coarse spaces for Helmholtz problems in the high frequency regime

Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for these methods is either out of reach with standard mathematical tools or does not cover all cases of practical interest. More precisely, we will focus on the comparison of three coarse spaces that yield two-level methods: the grid coarse space, DtN coarse space, and GenEO coarse space. We will show that they display different pros and cons, and properties depending o