Chapter 21
Solving Partial Differential Equations Using Finite Difference Method
Chapter 21 Solving Partial Differential Equations Using Finite Difference Method
by Fayez Gebali
Algorithms and Parallel Computing
- Cover
- Title page
- Copyright page
- Dedication
- Preface
- List of Acronyms
- Chapter 1 Introduction
- 1.1 INTRODUCTION
- 1.2 TOWARD AUTOMATING PARALLEL PROGRAMMING
- 1.3 ALGORITHMS
- 1.4 PARALLEL COMPUTING DESIGN CONSIDERATIONS
- 1.5 PARALLEL ALGORITHMS AND PARALLEL ARCHITECTURES
- 1.6 RELATING PARALLEL ALGORITHM AND PARALLEL ARCHITECTURE
- 1.7 IMPLEMENTATION OF ALGORITHMS: A TWO-SIDED PROBLEM
- 1.8 MEASURING BENEFITS OF PARALLEL COMPUTING
- 1.9 AMDAHL’S LAW FOR MULTIPROCESSOR SYSTEMS
- 1.10 GUSTAFSON–BARSIS’S LAW
- 1.11 APPLICATIONS OF PARALLEL COMPUTING
- Chapter 2 Enhancing Uniprocessor Performance
- Chapter 3 Parallel Computers
- 3.1 INTRODUCTION
- 3.2 PARALLEL COMPUTING
- 3.3 SHARED-MEMORY MULTIPROCESSORS (UNIFORM MEMORY ACCESS [UMA])
- 3.4 DISTRIBUTED-MEMORY MULTIPROCESSOR (NONUNIFORM MEMORY ACCESS [NUMA])
- 3.5 SIMD PROCESSORS
- 3.6 SYSTOLIC PROCESSORS
- 3.7 CLUSTER COMPUTING
- 3.8 GRID (CLOUD) COMPUTING
- 3.9 MULTICORE SYSTEMS
- 3.10 SM
- 3.11 COMMUNICATION BETWEEN PARALLEL PROCESSORS
- 3.12 SUMMARY OF PARALLEL ARCHITECTURES
- Chapter 4 Shared-Memory Multiprocessors
- Chapter 5 Interconnection Networks
- Chapter 6 Concurrency Platforms
- Chapter 7 Ad Hoc Techniques for Parallel Algorithms
- Chapter 8 Nonserial–Parallel Algorithms
- 8.1 INTRODUCTION
- 8.2 COMPARING DAG AND DCG ALGORITHMS
- 8.3 PARALLELIZING NSPA ALGORITHMS REPRESENTED BY A DAG
- 8.4 FORMAL TECHNIQUE FOR ANALYZING NSPAs
- 8.5 DETECTING CYCLES IN THE ALGORITHM
- 8.6 EXTRACTING SERIAL AND PARALLEL ALGORITHM PERFORMANCE PARAMETERS
- 8.7 USEFUL THEOREMS
- 8.8 PERFORMANCE OF SERIAL AND PARALLEL ALGORITHMS ON PARALLEL COMPUTERS
- Chapter 9 z-Transform Analysis
- 9.1 INTRODUCTION
- 9.2 DEFINITION OF z-TRANSFORM
- 9.3 THE 1-D FIR DIGITAL FILTER ALGORITHM
- 9.4 SOFTWARE AND HARDWARE IMPLEMENTATIONS OF THE z-TRANSFORM
- 9.5 DESIGN 1: USING HORNER’S RULE FOR BROADCAST INPUT AND PIPELINED OUTPUT
- 9.6 DESIGN 2: PIPELINED INPUT AND BROADCAST OUTPUT
- 9.7 DESIGN 3: PIPELINED INPUT AND OUTPUT
- Chapter 10 Dependence Graph Analysis
- 10.1 INTRODUCTION
- 10.2 THE 1-D FIR DIGITAL FILTER ALGORITHM
- 10.3 THE DEPENDENCE GRAPH OF AN ALGORITHM
- 10.4 DERIVING THE DEPENDENCE GRAPH FOR AN ALGORITHM
- 10.5 THE SCHEDULING FUNCTION FOR THE 1-D FIR FILTER
- 10.6 NODE PROJECTION OPERATION
- 10.7 NONLINEAR PROJECTION OPERATION
- 10.8 SOFTWARE AND HARDWARE IMPLEMENTATIONS OF THE DAG TECHNIQUE
- Chapter 11 Computational Geometry Analysis
- 11.1 INTRODUCTION
- 11.2 MATRIX MULTIPLICATION ALGORITHM
- 11.3 THE 3-D DEPENDENCE GRAPH AND COMPUTATION DOMAIN D
- 11.4 THE FACETS AND VERTICES OF D
- 11.5 THE DEPENDENCE MATRICES OF THE ALGORITHM VARIABLES
- 11.6 NULLSPACE OF DEPENDENCE MATRIX: THE BROADCAST SUBDOMAIN B
- 11.7 DESIGN SPACE EXPLORATION: CHOICE OF BROADCASTING VERSUS PIPELINING VARIABLES
- 11.8 DATA SCHEDULING
- 11.9 PROJECTION OPERATION USING THE LINEAR PROJECTION OPERATOR
- 11.10 EFFECT OF PROJECTION OPERATION ON DATA
- 11.11 THE RESULTING MULTITHREADED/MULTIPROCESSOR ARCHITECTURE
- 11.12 SUMMARY OF WORK DONE IN THIS CHAPTER
- Chapter 12 Case Study: One-Dimensional IIR Digital Filters
- Chapter 13 Case Study: Two- and Three-Dimensional Digital Filters
- Chapter 14 Case Study: Multirate Decimators and Interpolators
- 14.1 INTRODUCTION
- 14.2 DECIMATOR STRUCTURES
- 14.3 DECIMATOR DEPENDENCE GRAPH
- 14.4 DECIMATOR SCHEDULING
- 14.5 DECIMATOR DAG FOR s1 = [1 0]
- 14.6 DECIMATOR DAG FOR s2 = [1 −1]
- 14.7 DECIMATOR DAG FOR s3 = [1 1]
- 14.8 POLYPHASE DECIMATOR IMPLEMENTATIONS
- 14.9 INTERPOLATOR STRUCTURES
- 14.10 INTERPOLATOR DEPENDENCE GRAPH
- 14.11 INTERPOLATOR SCHEDULING
- 14.12 INTERPOLATOR DAG FOR s1 = [1 0]
- 14.13 INTERPOLATOR DAG FOR s2 = [1 −1]
- 14.14 INTERPOLATOR DAG FOR s3 = [1 1]
- 14.15 POLYPHASE INTERPOLATOR IMPLEMENTATIONS
- Chapter 15 Case Study: Pattern Matching
- 15.1 INTRODUCTION
- 15.2 EXPRESSING THE ALGORITHM AS A REGULAR ITERATIVE ALGORITHM (RIA)
- 15.3 OBTAINING THE ALGORITHM DEPENDENCE GRAPH
- 15.4 DATA SCHEDULING
- 15.5 DAG NODE PROJECTION
- 15.6 DESIGN 1: DESIGN SPACE EXPLORATION WHEN s = [1 1]t
- 15.7 DESIGN 2: DESIGN SPACE EXPLORATION WHEN s = [1 −1]t
- 15.8 DESIGN 3: DESIGN SPACE EXPLORATION WHEN s = [1 0]t
- Chapter 16 Case Study: Motion Estimation for Video Compression
- Chapter 17 Case Study: Multiplication over GF(2m)
- 17.1 INTRODUCTION
- 17.2 THE MULTIPLICATION ALGORITHM IN GF(2m)
- 17.3 EXPRESSING FIELD MULTIPLICATION AS AN RIA
- 17.4 FIELD MULTIPLICATION DEPENDENCE GRAPH
- 17.5 DATA SCHEDULING
- 17.6 DAG NODE PROJECTION
- 17.7 DESIGN 1: USING d1 = [1 0]t
- 17.8 DESIGN 2: USING d2 = [1 1]t
- 17.9 DESIGN 3: USING d3 = [1 −1]t
- 17.10 APPLICATIONS OF FINITE FIELD MULTIPLIERS
- Chapter 18 Case Study: Polynomial Division over GF(2)
- 18.1 INTRODUCTION
- 18.2 THE POLYNOMIAL DIVISION ALGORITHM
- 18.3 THE LFSR DEPENDENCE GRAPH
- 18.4 DATA SCHEDULING
- 18.5 DAG NODE PROJECTION
- 18.6 DESIGN 1: DESIGN SPACE EXPLORATION WHEN s1 = [1 −1]
- 18.7 DESIGN 2: DESIGN SPACE EXPLORATION WHEN s2 = [1 0]
- 18.8 DESIGN 3: DESIGN SPACE EXPLORATION WHEN s3 = [1 −0.5]
- 18.9 COMPARING THE THREE DESIGNS
- Chapter 19 The Fast Fourier Transform
- Chapter 20 Solving Systems of Linear Equations
- Chapter 21 Solving Partial Differential Equations Using Finite Difference Method
- References
- Index
-
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