Nonlinear multigrid for fully-implicit and high-order accurate simluation of multiphase flow in porous media

High-order accurate and fully implicit finite difference schemes are widely used for multiphase flow problems. In this paper we analyze the use of point Gauss-Seidel relaxation in a nonlinear multigrid method for the resulting nonlinear systems of equations. Point Gauss-Seidel is unstable for calculating the steady-state of high-order accurate discretizations of the 2D convection equation. Here we present a local Fourier mode smoothing analysis for the transient case. It appears that point Gauss-Seidel is a good smoother provided that the time step is taken small enough. Numerical computations show good multigrid convergence rates for typical test problems

A simple cell-centered multigrid method for 33D interface problems

AbstractA multigrid algorithm is presented for cell-centered discretizations of interface problems. Instead of constructing the coarse grid operators by means of the Galerkin approximation, the coarse grid operators are obtained by discretization on the coarse grids. The advantage of this approach is that we obtain M-matrices on all grids, and that the sparsity pattern of the fine grid matrix is retained on all grids. Moreover, the coarse grid operators are very easy to construct. Numerical results of several test problems are presented

Nonlinear multigrid for fully-implicit and high-order accurate simulation of multiphase flow in porous media

High-order accurate and fully implicit finite difference schemes are widely used for multiphase flow problems. In this paper we analyze the use of point Gauss-Seidel relaxation in a nonlinear multigrid method for the resulting nonlinear systems of equations. Point Gauss-Seidel is unstable for calculating the steady-state of high-order accurate discretizations of the 2D convection equation. Here we present a local Fourier mode smoothing analysis for the transient case. It appears that point Gauss-Seidel is a good smoother provided that the time step is taken small enough. Numerical computations show good multigrid convergence rates for typical test problems

Multigrid methods for high-order accurate fully implicit simulation of flowin porous media

High-order accurate finite difference schemes are widely used to avoid the detrimental effects of numerical diffusion in first-order upwind schemes. If an implicit time integration scheme is employed, we have to solve large systems of nonlinear equations in every time step. The price paid for the high-order accuracy is a larger discretization stencil. In this paper we consider the use of multigrid methods for the iterative solutions of these systems of equations. We consider both a direct multigrid approach and a defect correction approach, in which only first-order accurate discretized problems have to be solved. In both approaches we do not need to store the Jacobean matrix. Therefore the memory requirements are moderate, and very fine grid simulations are feasible on a standard workstation

Design patterns and software techniques for large-scale, open and reproducible data reduction

The preparation for the construction of the Square Kilometre Array, and the introduction of its operational precursors, such as LOFAR and MeerKAT, mark the beginning of an exciting era for astronomy. Impressive new data containing valuable science just waiting for discovery is already being generated, and these devices will produce far more data than has ever been collected before. However, with every new data instrument, the data rates grow to unprecedented quantities of data, requiring novel new data-processing tools. In addition, creating science grade data from the raw data still requires significant expert knowledge for processing this data. The software used is often developed by a scientist who lacks proper training in software development skills, resulting in the software not progressing beyond a prototype stage in quality. In the first chapter, we explore various organisational and technical approaches to address these issues by providing a historical overview of the development of radioastronomy pipelines since the inception of the field in the 1940s. In that, the steps required to create a radio image are investigated. We used the lessons-learned to identify patterns in the challenges experienced, and the solutions created to address these over the years. The second chapter describes the mathematical foundations that are essential for radio imaging. In the third chapter, we discuss the production of the KERN Linux distribution, which is a set of software packages containing most radio astronomy software currently in use. Considerable effort was put into making sure that the contained software installs appropriately, all items next to one other on the same system. Where required and possible, bugs and portability fixes were solved and reported with the upstream maintainers. The KERN project also has a website, and issue tracker, where users can report bugs and maintainers can coordinate the packaging effort and new releases. The software packages can be used inside Docker and Singularity containers, enabling the installation of these packages on a wide variety of platforms. In the fourth and fifth chapters, we discuss methods and frameworks for combining the available data reduction tools into recomposable pipelines and introduce the Kliko specification and software. This framework was created to enable end-user astronomers to chain and containerise operations of software in KERN packages. Next, we discuss the Common Workflow Language (CommonWL), a similar but more advanced and mature pipeline framework invented by bio-informatics scientists. CommonWL is supported by a wide range of tools already; among other schedulers, visualisers and editors. Consequently, when a pipeline is made with CommonWL, it can be deployed and manipulated with a wide range of tools. In the final chapter, we attempt something unconventional, applying a generative adversarial network based on deep learning techniques to perform the task of sky brightness reconstruction. Since deep learning methods often require a large number of training samples, we constructed a CommonWL simulation pipeline for creating dirty images and corresponding sky models. This simulated dataset has been made publicly available as the ASTRODECONV2019 dataset. It is shown that this method is useful to perform the restoration and matches the performance of a single clean cycle. In addition, we incorporated domain knowledge by adding the point spread function to the network and by utilising a custom loss function during training. Although it was not possible to improve the cleaning performance of commonly used existing tools, the computational time performance of the approach looks very promising. We suggest that a smaller scope should be the starting point for further studies and optimising of the training of the neural network could produce the desired results