Domain decomposition methods for the parallel computation of reacting flows

Domain decomposition is a natural route to parallel computing for partial differential equation solvers. Subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, comparisons are made between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demonstrate for it approximately 10-fold speedup on 16 processors

Finding apparent horizons in numerical relativity

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced H(h) function \H(\h) by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing \H(\h). Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum H(h) equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the \H(\h) Jacobian to be {\em much\/} more efficient than the usual numerical-perturbation method, and also much easier to implement than is commonly thought. When solving the (discrete) \H(\h) = 0 equations, we find that Newton's method generally converges very rapidly, although there are difficulties when the initial guess contains high-spatial-frequency errors. Using 4th~order finite differencing, we find typical accuracies for the horizon position in the 10^{-5} range for \Delta \theta = \frac{\pi/2}{50}

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Computational fluid dynamics

An overview of computational fluid dynamics (CFD) activities at the Langley Research Center is given. The role of supercomputers in CFD research, algorithm development, multigrid approaches to computational fluid flows, aerodynamics computer programs, computational grid generation, turbulence research, and studies of rarefied gas flows are among the topics that are briefly surveyed

Wavelet-based semiconductor device simulation.

by Pun Kong-Pang.Thesis (M.Phil.)--Chinese University of Hong Kong, 1997.Includes bibliographical references (leaves 94-[96]).Acknowledgement --- p.iAbstract --- p.iiiList of Tables --- p.viiList of Figures --- p.viiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Role of Device Simulation --- p.2Chapter 1.2 --- Classification of Device Models --- p.3Chapter 1.3 --- Sections of a Typical Simulator --- p.6Chapter 1.4 --- Arrangement of This Thesis --- p.7Chapter 2 --- Classical Physical Model --- p.9Chapter 2.1 --- Carrier Densities --- p.12Chapter 2.2 --- Space Charge --- p.14Chapter 2.3 --- Carrier Mobilities --- p.15Chapter 2.4 --- Generation and Recombination --- p.17Chapter 2.5 --- Modeling of Device Boundaries --- p.20Chapter 2.6 --- Limits of Classical Device Modeling --- p.22Chapter 3 --- Computational Aspects --- p.23Chapter 3.1 --- Normalization --- p.24Chapter 3.2 --- Discretization --- p.26Chapter 3.2.1 --- Finite Difference Method --- p.26Chapter 3.2.2 --- Finite Element Method --- p.27Chapter 3.3 --- Nonlinear Systems --- p.28Chapter 3.3.1 --- Newton's Method --- p.28Chapter 3.3.2 --- Gummel's Method and its modification --- p.29Chapter 3.3.3 --- Comparison and discussion --- p.30Chapter 3.4 --- Linear System and Sparse Matrix --- p.32Chapter 4 --- Cubic Spline Wavelet Collocation Method for PDEs --- p.34Chapter 4.1 --- Cubic spline scaling functions and wavelets --- p.35Chapter 4.1.1 --- Approximation for a function in H2(I) --- p.43Chapter 4.2 --- Wavelet interpolation --- p.45Chapter 4.2.1 --- Interpolant operator Ivo in Vo --- p.45Chapter 4.2.2 --- Interpolation operator IWjf in Wj --- p.47Chapter 4.3 --- Derivative Matrices --- p.51Chapter 4.3.1 --- First derivative matrix --- p.51Chapter 4.3.2 --- Second derivative matrix --- p.53Chapter 4.4 --- Wavelet Collocation Method for Solving Device Equations --- p.55Chapter 4.4.1 --- Steady state solution --- p.57Chapter 4.4.2 --- Transient solution --- p.58Chapter 4.5 --- Reducing Collocation Points --- p.59Chapter 4.5.1 --- Error evaluation --- p.59Chapter 4.5.2 --- Deleting collocation points --- p.61Chapter 5 --- Numerical Results --- p.64Chapter 5.1 --- P-N Junction Diode --- p.64Chapter 5.1.1 --- Steady state solution --- p.69Chapter 5.1.2 --- Transient solution --- p.76Chapter 5.1.3 --- Convergence --- p.79Chapter 5.2 --- Bipolar Transistor --- p.81Chapter 5.2.1 --- Boundary Model --- p.82Chapter 5.2.2 --- DC Solution --- p.83Chapter 5.2.3 --- Transient Solution --- p.89Chapter 6 --- Conclusions --- p.92Bibliography --- p.9

Optimization governed by stochastic partial differential equations

This thesis provides a rigorous framework for the solution of stochastic elliptic partial differential equation (SPDE) constrained optimization problems. In modeling physical processes with differential equations, much of the input data is uncertain (e.g. measurement errors in the diffusivity coefficients). When uncertainty is present, the governing equations become a family of equations indexed by a stochastic variable. Since solutions of these SPDEs enter the objective function, the objective function usually involves statistical moments. These optimization problems governed by SPDEs are posed as a particular class of optimization problems in Banach spaces. This thesis discusses Monte Carlo, stochastic Galerkin, and stochastic collocation methods for the numerical solution of SPDEs and identifies the stochastic collocation method as particularly useful for the optimization of SPDEs. This thesis extends the stochastic collocation method to the optimization context and explores the decoupling nature of this method for gradient and Hessian computations

Institute for Computer Applications in Science and Engineering (ICASE)

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science during the period April 1, 1983 through September 30, 1983 is summarized