Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel??s one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel??s basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1991.

Second order level set projection method for incompressible Navier-Stokes equations is proposed to solve flow around arbitrary geometries. We used rectilinear grid with collocated cell centered velocity and pressure. An explicit Godunov procedure is used to address the nonlinear advection terms, and an implicit Crank-Nicholson method to update viscous effects. An approximate pressure projection is implemented at the end of the time stepping using multigrid as a conventional fast iterative method. The level set method developed by Osher and Sethian [17] is implemented to address real momentum and pressure boundary conditions by the advection of a distance function, as proposed by Aslam [3]. Numerical results for the Strouhal number and drag coefficients validated the model with good accuracy for flow over a cylinder in the parallel shedding regime (47 < Re < 180). Simulations for an array of cylinders and an oscillating cylinder were performed, with the latter demonstrating our methods ability to handle dynamic boundary conditions.

O objetivo deste trabalho é o estudo de métodos multigrid para a solução de equações elípticas na esfera, discretizadas em malhas esféricas geodésicas icosaédricas. Malhas esféricas geradas a partir de sólidos platônicos receberam crescente atenção ao longo da última década, por serem razoavelmente uniformes e não apresentarem concentração de pontos em torno dos pólos como as tradicionais malhas latitude-longitude. Em especial, as malhas geodésicas icosaédricas (geradas a partir de um icosaedro inscrito na esfera com suas faces projetadas na superfície) têm sido adotadas no desenvolvimento de diversos modelos atmosféricos. Nestes é comum a necessidade de resolução de equações do tipo Poisson como parte do método de integração, motivando o nosso trabalho. Adotamos uma discretização do operador de Laplace baseada em volumes finitos. Para tal escrevemos o laplaciano como o divergente do gradiente. O divergente é discretizado com base nos fluxos nos pontos médios das arestas das células computacionais (com o auxílio do teorema da divergência de Gauss) e no uso de diferenças centradas para aproximar as derivadas nesses pontos médios. Validamos a discretização para o operador de Laplace resolvendo uma equação de Poisson através dos métodos iterativos de Jacobi e Gauss-Seidel. Estes sabidamente não são eficientes computacionalmente, devido ao grande e crescente número de iterações necessárias para atingir a convergência ao refinar a malha. Uma alternativa muito eficiente para a resolução de equações elípticas é a métodologia multigrid. Investigamos alguns métodos multigrid propostos na literatura para a solução destas equações na malha esférica geodésica icosaédrica. A partir desse estudo, utilizando também como referência a Análise Local de Fourier para a equação de Poisson em malhas hexagonais uniformes, como uma aproximação para malhas geodésicas icosaédricas, escolhemos um algoritmo multigrid para implementação. Testamos algumas opções para as componentes do esquema multigrid. Obtivemos taxas de convergência muito boas com V(1,1) ciclos com relaxação por Gauss-Seidel, restrição full weighting e interpolação linear. / This work is dedicated to the numerical solution of elliptic equations on the sphere, discretized on geodesic icosahedral grids. Spherical meshes generated from projections of platonic solids received considerable attention in the last decade, once they are almost isotropic and do not present a concentration of grid points around the poles, as traditional latitude-longitude grids. In particular, the geodesic icosahedral spherical grids have been adopted in the development of several atmospheric models. In these models, the necessity to solve Poisson type equations is very common, providing a motivation for our present work. We have employed a discretization of the Laplace operator based on finite volumes. We write the Laplacian as the divergent of the gradient operator and use Gauss theorem to derive the discretization of the operator. We integrate the fluxes along the cell borders and approximate them through finite-differences. We first validated the discretization solving Poisson\'s equation with a simple (and very innefficient) Jacobi-Relaxation and Gauss-Seidel. We then investigated the use of multigrid type schemes for the solution of this equation. We have analysed some schemes proposed in the literature, also using an idealized Local Fourier Analysis on hexagonal (planar) grids to estimate the behaviour of the schemes on the icosaedral grids. We have implemented and tested a multigrid method, comparing the performance with different relaxation schemes and transfer operators. We have obtained a very efficient method employing V(1,1) cycles with Gauss-Seidel relaxation, and full-weighting and linear interpolation as transfer-operators.

O objetivo deste trabalho é o estudo de métodos multigrid para a solução de equações elípticas na esfera, discretizadas em malhas esféricas geodésicas icosaédricas. Malhas esféricas geradas a partir de sólidos platônicos receberam crescente atenção ao longo da última década, por serem razoavelmente uniformes e não apresentarem concentração de pontos em torno dos pólos como as tradicionais malhas latitude-longitude. Em especial, as malhas geodésicas icosaédricas (geradas a partir de um icosaedro inscrito na esfera com suas faces projetadas na superfície) têm sido adotadas no desenvolvimento de diversos modelos atmosféricos. Nestes é comum a necessidade de resolução de equações do tipo Poisson como parte do método de integração, motivando o nosso trabalho. Adotamos uma discretização do operador de Laplace baseada em volumes finitos. Para tal escrevemos o laplaciano como o divergente do gradiente. O divergente é discretizado com base nos fluxos nos pontos médios das arestas das células computacionais (com o auxílio do teorema da divergência de Gauss) e no uso de diferenças centradas para aproximar as derivadas nesses pontos médios. Validamos a discretização para o operador de Laplace resolvendo uma equação de Poisson através dos métodos iterativos de Jacobi e Gauss-Seidel. Estes sabidamente não são eficientes computacionalmente, devido ao grande e crescente número de iterações necessárias para atingir a convergência ao refinar a malha. Uma alternativa muito eficiente para a resolução de equações elípticas é a métodologia multigrid. Investigamos alguns métodos multigrid propostos na literatura para a solução destas equações na malha esférica geodésica icosaédrica. A partir desse estudo, utilizando também como referência a Análise Local de Fourier para a equação de Poisson em malhas hexagonais uniformes, como uma aproximação para malhas geodésicas icosaédricas, escolhemos um algoritmo multigrid para implementação. Testamos algumas opções para as componentes do esquema multigrid. Obtivemos taxas de convergência muito boas com V(1,1) ciclos com relaxação por Gauss-Seidel, restrição full weighting e interpolação linear. / This work is dedicated to the numerical solution of elliptic equations on the sphere, discretized on geodesic icosahedral grids. Spherical meshes generated from projections of platonic solids received considerable attention in the last decade, once they are almost isotropic and do not present a concentration of grid points around the poles, as traditional latitude-longitude grids. In particular, the geodesic icosahedral spherical grids have been adopted in the development of several atmospheric models. In these models, the necessity to solve Poisson type equations is very common, providing a motivation for our present work. We have employed a discretization of the Laplace operator based on finite volumes. We write the Laplacian as the divergent of the gradient operator and use Gauss theorem to derive the discretization of the operator. We integrate the fluxes along the cell borders and approximate them through finite-differences. We first validated the discretization solving Poisson\'s equation with a simple (and very innefficient) Jacobi-Relaxation and Gauss-Seidel. We then investigated the use of multigrid type schemes for the solution of this equation. We have analysed some schemes proposed in the literature, also using an idealized Local Fourier Analysis on hexagonal (planar) grids to estimate the behaviour of the schemes on the icosaedral grids. We have implemented and tested a multigrid method, comparing the performance with different relaxation schemes and transfer operators. We have obtained a very efficient method employing V(1,1) cycles with Gauss-Seidel relaxation, and full-weighting and linear interpolation as transfer-operators.

In the recent past, the field of scientific computing has become of more and more importance for scientific as well as for industrial research, playing a comparable role as experiment and theory do. This success of computational methods in scientific and engineering research is next to the enormous improvement of computer hardware to a large extend due to contributions from applied mathematicians, who have developed algorithms which make real life applications feasible. Examples are adaptive methods, high order discretization, fast linear and non-linear solvers and multi-level methods. The application of these methods in a large class of problems demands for suitable and robust tools for a flexible and efficient implementation. In order to play a crucial role in scientific and engineering research, besides efficiency in the numerical solution, also efficiency in problem setup and interpretation of simulation results is of utmost importance. As modeling and computing comes closer together, efficient computational methods need to be applied to new sets of equations. The problems to be addressed by simulation methods become more and more complicated, ranging over different scales, interacting on different dimensions and combining different physics. Such problems need to be implemented in a short period of time, solved on complicated domains and visualized with respect to the demand of the user. %Only a modular abstract simulation environment will fulfill these requirements and allow to setup, solve and visualize real-world problems appropriately. In this work, the concepts and the design of the C++ finite element toolbox AMDiS (adaptive multidimensional simulations) are described. It is shown, how abstract data structures and modern software concepts can help to design user-friendly finite element software, which provides large flexibility in problem definition while on the other hand efficiently solves these problems. Also systems of coupled problems can be solved in an intuitive way. In order to demonstrate its possibilities, AMDiS has been applied to several non-standard problems. The most time-consuming part in most simulations is the solution of linear systems of equations. Multi-level methods use discretization hierarchies to solve these systems in a very efficient way. In AMDiS, such multi-level techniques are implemented in the context of adaptive finite elements. Several numerical results are given which compare this multigrid solver with classical iterative methods. Besides the development of more efficient algorithms also the growing hardware capabilities lead to an improvement of simulation possibilities. Modern computing clusters contain more and more processors and also personal computers today are often equipped with multi-core processors. In this work, a new parallelization approach has been developed which allows the parallelization of sequential code in a very easy way and reduces the communication overhead compared to classical parallelization concepts.

The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required.

The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required.

O objetivo principal deste trabalho é a paralelização de um modelo global de previsão do tempo em diferenças finitas com refinamento local. Este é baseado nas equações primitivas, e faz uso de uma discretização semi-Lagrangiana e semi-implícita em três níveis no tempo em uma malha de Lorenz na vertical e uma malha do tipo C de Arakawa na horizontal. A discretização horizontal é feita através de diferenças finitas de segunda ordem. A equação escalar elíptica tridimensional resultante é desacoplada em um sistema de equações bidimensionais do tipo Helmholtz, o qual é resolvido por meio de um método multigrid. O modelo de paralelização foi desenvolvido para máquinas com memória distribuída, fazendo uso de MPI para passagens de mensagens e baseado em técnicas de decomposição de domínio. O acoplamento apenas local dos operadores de diferenças finitas viabiliza a decomposição em duas direções horizontais. Evitamos a decomposição vertical, tendo em vista o forte acoplamento nesta direção das parametrizações de fenômenos físicos. A estratégia de paralelização foi elaborada visando o uso eficiente de centenas ou alguns milhares de processadores, dependendo da resolução do modelo. Para tal, a malha localmente refinada é separada em três regiões: uma grossa, uma de transição e uma fina, onde cada uma delas é dividida de forma independente entre um número de processadores proporcional ao número de pontos que cada uma armazena, garantindo assim um balanceamento de carga adequado. Não obstante, para resolver o sistema de equações bidimensionais do tipo Helmholtz foi necessário mudar a estratégia de paralelização, dividindo o domínio unicamente nas direções vertical e latitudinal. Ambas partes do modelo com paralelizações diferentes estão conectadas por meio da estratégia de transposição de dados. Testamos nosso modelo utilizando até 1024 processadores e os resultados ainda mostraram uma boa escalabilidade. / The main goal of this work is the parallelization of a weather prediction model employing finite differences on locally refined meshes. The model is based on the primitive equations and uses a three-time-level semi-implicit semi-Lagrangian temporal discretization on a Lorenz-type vertical grid combined with a horizontal Arakawa C-grid. The horizontal discretization is performed by means of second order finite differences. The resulting three-dimensional scalar elliptic equation is decoupled into a set of Helmholtz-type two-dimensional equations, solved by a multigrid method. The parallelization has been written for distributed-memory machines, employing the MPI message passing standard and was based on domain decomposition techniques. The local coupling of the finite difference operators was exploited in a two-dimensional horizontal decomposition. We avoid a vertical decomposition due to the strong coupling of physical parameterization routines. The parallelization strategy has been designed in order to allow the efficient use of hundreds to a few thousand processors, depending on the model resolution. In order to achieve this, the locally refined mesh is split into three regions: a coarse, a transition and a fine one, each decomposed independently. The number of allocated processors for each region is proportional to the number of the grid-points it contains, in order to guarantee a good load-balancing distribution. However, to solve the set of Helmholtz-type bidimensional equations it was necessary to change the parallelization strategy, splitting the domain only in vertical and latitudinal directions. Both parts of the model with different parallelizations are related by means the data transposition strategy. We tested our model using up to 1024 processors and the results still showed a good scalability.