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Elliptic Marching Methods and Domain Decomposition

Patrick J. Roache



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CRC Press Inc
29 June 1995
One of the first things a student of partial differential equations learns is that it is impossible to solve elliptic equations by spatial marching. This new book describes how to do exactly that, providing a powerful tool for solving problems in fluid dynamics, heat transfer, electrostatics, and other fields characterized by discretized partial differential equations. Elliptic Marching Methods and Domain Decomposition demonstrates how to handle numerical instabilities (i.e., limitations on the size of the problem) that appear when one tries to solve these discretized equations with marching methods. The book also shows how marching methods can be superior to multigrid and pre-conditioned conjugate gradient (PCG) methods, particularly when used in the context of multiprocessor parallel computers. Techniques for using domain decomposition together with marching methods are detailed, clearly illustrating the benefits of these techniques for applications in engineering, applied mathematics, and the physical sciences.
By:   Patrick J. Roache
Imprint:   CRC Press Inc
Country of Publication:   United States
Volume:   5
Dimensions:   Height: 254mm,  Width: 178mm,  Spine: 16mm
Weight:   567g
ISBN:   9780849373787
ISBN 10:   0849373786
Series:   Symbolic & Numeric Computation
Pages:   208
Publication Date:   29 June 1995
Audience:   College/higher education ,  Professional and scholarly ,  Further / Higher Education ,  Undergraduate
Format:   Hardback
Publisher's Status:   Active
Basic Marching Methods for 2D Elliptic Problems Introduction: The Impact of Direct Methods. Direct Marching Methods. History of Marching Methods. The Marching Method in 1D. The Reference 2D Problem. Operation Counts as an Index of Merit. Operation Counts for the Reference 2D Problem. Error Propagation Characteristics for the Reference 2D Problem. Gradient, Mixed, and Periodic Boundary Conditions. Irregular Mesh and Variable Coefficient Poisson Equations. Irregular Geometries. Other Second-Order Elliptic Equations: Advection-Diffusion Equations. Upwind Differences. Turbulence Terms. Fibonacci Scale. Helmholtz Terms. Cross Derivatives. Gradient Boundary Conditions and Cross Derivatives. Interior Flux Boundaries. High-Order Equations Introduction High-Order Accuracy Operators Higher-Order Accurate Solutions by Deferred Corrections Higher-Order Elliptic Equations Operation Counts for Higher-Order Systems Finite Element Equations Extending the Mesh Size: Domain Decomposition Introduction Mesh Doubling by Two-Directional Marching Multiple Marching Patching Influence Extending Other Direct Methods for Extending the Mesh Size Lower Accuracy Stencils Plus Iteration Iterative Coupling for Subregions Higher Precision Arithmetic: Applications on Workstations and Virtual Parallel Networks Banded Approximations to Influence Matrices Introduction Banded Approximation to C Operation Count and Storage for Banded CB Intrinsic Storage: Data Compression for Massively Parallel Computers Banded Approximation to C (*****Need a hat or caret over the C in the above title) Marching Methods in 3D Introduction Simple 3D Marching Error Propagation Characteristics for the 3D EVP Operation Count and Storage Penalty for the 3D EVP Method Banded Approximations in 3D Operation Count for Banded Approximation in 3D Additional Terms in the 3D Marching Method 3D EVP-FFT Method Error Propagation Characteristics for 3D EVP-FFT Method Operation Count and Storage Penalty for the 3D EVP-FFT Method Accuracy and Additional Terms in the 3D EVP-FFT Method N-Plane Relaxation within Multigrid and Domain Decomposition Methods Performance of the 2D GEM Code Introduction Uses and Users Overview of the GEM Codes Problem Description in the Basic GEM Code Tests of the Basic GEM Code The Stabilizing Codes GEMPAT2 and GEMPAT4 Timing Tests of the Stabilized Codes Representative Accuracy Testing Conclusions Vectorization and Parallelization Introduction Vectorizing the Tridiagonal Algorithm and the 9-Point March Vectorizing the 5-Point March Timing and Accuracy for the Vectorized Marches Efficiencies Multiprocessor Architectures Semidirect Methods for Nonlinear Equations of Fluid Dynamics Introduction: Time-Dependent Calculations vs. Semidirect Methods Burgers Equation by Time Accurate Methods Basic Idea of Semidirect Methods Burgers Equation by Picard Semidirect Iteration Further Discussion of the Picard Semidirect Iteration Genesis of Semidirect Methods NOS Method LAD Method Performance of NOS and LAD on the Driven Cavity Problem Relative Importance of Lagging Boundary Conditions Performance of LAD and NOS on a Flow-Through Problem Optimum Relaxation Factor and Convergence for Large Problems Choice Between LAD and NOS Split NOS Method A Better Boundary Condition on Wall Vorticity Dorodnicyn-Meller Method Viscous Flows in Alternate Variables BID Method FOD and Coupled System Solvers Other Applications and Non-Time-Like Methods Remarks on Solution Uniqueness Remarks on Semidirect Methods within Domain Decomposition Comparison to Multigrid Methods Introduction Definition of the Methods Treatment of Nonlinearities Speed and Accuracy Grid Sensitivity and Word Length Sensitivity Directionality Storage Penalty Dimensionality Work Estimates Boundary Conditions General Coefficient Problems Grid Transformations Irregular Logical-Space Geometry Higher Order Systems Higher Order Accuracy Equations Finite Element Equations Use in Time Dependent Problems Cell Reynolds Number Difficulties Virtual Problems MLAT and Other Grid Adaptation Vectorization, Parallelization, and Convergence Testing Simplicity, Modularity, and Robustness Summary Appendix A - Marching Schemes and Error Propagation for Various Discrete Laplacians Appendix B - Tridiagonal Algorithm for Periodic Boundary Conditions Appendix C - Gauss Elimination as a Direct Solver Subject Index Each Chapter and Appendix also Contains a List of References

Reviews for Elliptic Marching Methods and Domain Decomposition

Together with an important historical perspective, this book uses the domain decomposition connection to develop and explore the nature of marching methods. Interesting analytical and anecdotal comparisons are made with direct methods and multigrid techniques, told by a scientist who has obviously has much experience with real practical problems. -Mathematical Reviews, 99a

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