Control of spatial discretisation in coastal oil spill modelling

Spatial discretisation plays an important role in many numerical environmental models. This paper studies the control of spatial discretisation in coastal oil spill modelling with a view to assure the quality of modelling outputs for given spatial data inputs. Spatial data analysis techniques are effective for investigating and improving the spatial discretisation in different phases of the modelling. Proposed methods are implemented and tested with experimental models. A new “Automatic Search” method based on GIS zone design principles is shown to significantly improve discretisation of bathymetric data and hydrodynamic modelling outputs. The concepts and methods developed in the study are expected to have general relevance for a range of applications in numerical environmental modelling

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth