Elliptic Marching Methods and Domain Decomposition: 1st Edition (Hardback) book cover

Elliptic Marching Methods and Domain Decomposition

1st Edition

By Patrick J. Roache

CRC Press

208 pages

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Hardback: 9780849373787
pub: 1995-06-29
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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.


"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

Table of Contents

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


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


Mesh Doubling by Two-Directional Marching

Multiple Marching


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


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


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


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


Vectorization and Parallelization


Vectorizing the Tridiagonal Algorithm and the 9-Point March

Vectorizing the 5-Point March

Timing and Accuracy for the Vectorized Marches


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


Definition of the Methods

Treatment of Nonlinearities

Speed and Accuracy

Grid Sensitivity and Word Length Sensitivity


Storage Penalty


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


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

About the Series

Symbolic & Numeric Computation

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations