Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions

Discontinuous Galerkin methods for solving the acoustic wave equation

In this work we develop a numerical simulator for the propagation of elastic waves by solving the one-dimensional acoustic wave equation with Absorbing Boundary Conditions (ABC’s) on the computational boundaries using Discontinuous Galerkin Finite Element Methods (DGFEM). The DGFEM allows us to easily simulate the presence of a fracture in the elastic medium by means of a linear-slip model. We analize the behaviour of our algorithm by comparing its results against analytic solutions. Furthermore, we show the frequency-dependent effect on the propagation produced by the fracture as appears in previous works. Finally, we present an analysis of the numerical parameters of the method.Fil: Castromán, Gabriel Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de la Plata. Facultad de Ciencias Astronómicas y Geofísicas. Departamento de Geofísica Aplicada; ArgentinaFil: Zyserman, Fabio Ivan. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de la Plata. Facultad de Ciencias Astronómicas y Geofísicas. Departamento de Geofísica Aplicada; Argentin

Computational and numerical analysis of differential equations using spectral based collocation method.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods, both modified and new, and apply them to solve differential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of differential equations defined over simple geometries. Procedurally, these methods entail transforming the gov erning differential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deficiency, the present study explores the development of numerical algorithms for solving ordinary and partial differential equations defined over simple geometries. The solutions of the differential equations considered are approximated using interpolating polynomials that satisfy the given differential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve differential equations defined on large computational domains much effort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con firms the superiority of these multiple domain techniques in terms of accuracy and computational efficiency over the single domain approach when applied to problems defined over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across different dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary differential equations. Subsequent chapters then build on solutions to partial differential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical algorithms. The numerical results of the study confirm that incorporating domain decomposition techniques in spectral collocation methods work effectively for all dimensions, as we report highly accurate results obtained in a computationally efficient manner for problems defined on large do mains. The findings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter

Mathematical software for gas transmission networks

This thesis is concerned with the development of numerical software for the simulation of gas transmission networks. This involves developing software for the solution of a large system of stiff differential/algebraic equations (DAE) containing frequent severe disturbances. The disturbances arise due to the varying consumer demands and the operation of network controlling devices such as the compressors. Special strategies are developed to solve the DAE system efficiently using a variable-step integrator. Two sets of strategies are devised; one for the implicit methods such as the semi-implicit Runge-Kutta method, and the other for the linearly implicit Rosenbrock-type method. Four integrators, based on different numerical methods, have been implemented and the performance of each one is compared with the British Gas network analysis program PAN, using a number of large, realistic transmission networks. The results demonstrate that the variable-step integrators are reliable and efficient. An efficient sparse matrix decomposition scheme is developed to solve the large, sparse system of equations that arise during the integration of the DAE system. The decomposition scheme fully exploits the special structure of the coefficient matrix. Lastly, for certain networks, the existing simulation programs fail to compute a feasible solution because of the interactions of the controlling devices in the network. To overcome this difficulty, the problem is formulated as a variational inequality model and solved numerically using an optimization routine from the NAG library (NAGFLIB(l982)). The reliability of the model is illustrated using three test networks

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above

A Class of Stable, Globally Noniterative, Nonoverlapping Domain Decomposition Algorithms for the Simulation of Parabolic Evolutionary Systems.

Parabolic systems are governed by time dependent partial differential equations. To obtain a high simulation quality that captures important features of a parabolic system requires solving the governing equation to an adequate accuracy, which necessitates a large sampling size in the spatial and temporal dimensions, and hence a large amount of simulation data and high computing cost. Domain decomposition is an effective method of divide-and-conquer paradigm that divides the problem domain into several subdomains, reducing the original problem into several smaller interdependent problems which can be solved in parallel. In this dissertation, we propose a class of stabilized explicit-implicit time marching (SEITM) domain decomposition algorithms for parabolic equations. Explicit-implicit time marching (EITM) algorithms are globally non-iterative nonoverlapping domain decomposition methods, which, when compared with Schwartz algorithm based parabolic solvers, are both computationally and communicationally efficient for each time step simulation but suffer from small time step size restrictions due to conditional stability. The proposed stabilization techniques in the SEITM algorithms retain the time-stepwise efficiency in computation and communication of the EITM algorithms but free the algorithms from small time step size restrictions, rendering SEITM algorithms excellent candidates for large scale parallel simulation problems. Three algorithms of the SEITM class are presented in this dissertation, which are mathematically analyzed and experimentally tested to show excellent numerical stability, computation and communication efficiencies, and high parallel speedup and scalability

Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd