Contents of Part 1: Finite Difference
Methods
0 Prelude
1 Introduction to Finite Differences
1.1 Introduction
1.2 Getting Started
1.2.1 Implementation
1.3 Consistency
1.3.1 Special Choice of 
and 
1.4 Neumann Boundary Conditions
1.5 Some Variations
1.5.1 Lower Order Terms
1.5.2 Nonhomogeneous Equations and Boundary Conditions
1.5.3 A Higher Order Scheme
1.6 Derivation of Difference Equations
1.6.1 Neumann Boundary Conditions
1.6.2 Cell Averaged Equations
1.6.3 Cell Centered Grids
1.6.4 Nonuniform Grids
2 Some Theoretical Considerations
2.1 Introduction
2.2 Convergence
2.2.1 Initial-Value Problems
2.2.2 Initial-Boundary-Value Problems
2.2.3 A Review of Linear Algebra
2.2.4 Some Additional Convergence Topics
2.3 Consistency
2.3.1 Initial-Value Problems
2.3.2 Initial-Boundary-Value Problems
2.4 Stability
2.4.1 Initial-Value Problems
2.4.2 Initial-Boundary-Value Problems
2.5 The Lax Theorem
2.5.1 Initial-Value Problems
2.5.2 Initial-Boundary-Value Problems
2.6 Computational Interlude I
2.6.1 Review of Computational Results
2.6.2 HW0.0.1
2.6.3 Implicit Schemes
2.6.4 Neumann Boundary Conditions
2.6.5 Derivation of Implicit Schemes
3 Stability
3.1 Analysis of Stability
3.1.1 Initial-Value Problems
3.1.2 Initial-Boundary-Value Problems
3.2 Finite Fourier Series and Stability
3.3 Gerschgorin Circle Theorem
3.4 Computational Interlude II
3.4.1 Review of Computational Results
3.4.2 HW0.0.1
4 Parabolic Equations
4.1 Introduction
4.2 Two Dimensional Parabolic Equations
4.2.1 Neumann Boundary Conditions
4.2.2 Derivation of Difference Equations
4.3 Convergence, Consistency, Stability
4.3.1 Stability of Initial-Value Schemes
4.3.2 Stability of Initial-Boundary-Value Schemes
4.4 Alternating Direction Implicit Schemes
4.4.1 Peaceman-Rachford Scheme
4.4.2 Initial-Value Problems
4.4.3 Initial-Boundary-Value Problems
4.4.4 Douglas-Rachford Scheme
4.4.5 Nonhomogeneous ADI Schemes
4.4.6 Three Dimensional Schemes
4.5 Polar Coordinates
5 Hyperbolic Equations
5.1 Introduction
5.2 Initial-Value Problems
5.3 Numerical Solution of Initial-Value Problems
5.3.1 One Sided Schemes
5.3.2 Centered Scheme
5.3.3 Lax-Wendroff Scheme
5.3.4 More Explicit Schemes
5.4 Implicit Schemes
5.4.1 One Sided Schemes
5.4.2 Centered Scheme
5.4.3 Lax-Wendroff Scheme
5.4.4 Crank-Nicolson Scheme
5.5 Initial-Boundary-Value Problems
5.5.1 Periodic Boundary Conditions
5.5.2 Dirichlet Boundary Conditions
5.6 Numerical Solution of Initial-Boundary-Value Problems
5.6.1 Periodic Boundary Conditions
5.6.2 Dirichlet Boundary Conditions
5.7 The Courant-Friedrichs-Lewy Condition
5.8 Two Dimensional Hyperbolic Equations
5.8.1 Conservation Law Derivation
5.8.2 Initial-Value Problems
5.8.3 ADI Schemes
5.8.4 Courant-Friedrichs-Lewy Condition for
Two Dimensional Problems
5.8.5 Two Dimensional
Initial-Boundary-Value Problems
5.9 Computational Interlude III
5.9.1 Review of Computational Results
5.9.2 Convection-Diffusion Equations
5.9.3 HW0.0.1
5.9.4 HW0.0.2
6 Systms of Partial Differential Equations
6.1 Introduction
6.2 Initial-Value Difference Schemes
6.2.1 Flux Splitting
6.2.2 Implicit Schemes
6.3 Initial-Boundary-Value Problems
6.3.1 Boundary Conditions
6.3.2 Implementation
6.4 Multilevel Schemes
6.4.1 Scalar Multilevel Schemes
6.4.2 Implementation of Scalar Multilevel Schemes
6.4.3 Multilevel Systems
6.5 Higher Order Hyperbolic Equations
6.5.1 Initial-Value Problems
6.5.2 More
6.6 Courant-Friedrichs-Lewy Condition for Systems
6.7 Two Dimensional Systems
6.7.1 Initial-Value Problems
6.7.2 Boundary Conditions
6.7.3 Two Dimensional Multilevel Schemes
6.8 A Consistent, Convergent, Unstable Difference Scheme?
6.9 Computational Interlude IV
6.9.1 HW0.0.1 and HW0.0.2
6.9.2 HW0.0.3
6.9.3 Parabolic Problems in Polar Coordinates
6.9.4 An Alternate Scheme for Polar Coordinates
7 Dispersion and Dissipation
7.1 Introduction
7.1.1 HW5.6.3
7.1.2 HW5.6.5
7.2 Dispersion and Dissipation for Partial Differential Equations
7.3 Dispersion and Dissipation for Difference Equations
7.4 Dispersion Analysis for the Leapfrog Scheme
7.5 More Dissipation
7.6 Artificial Dissipation
7.7 Modified Partial Differential Equation
7.8 Discontinuous Solutions
7.9 Computational Interlude V
7.9.1 HW0.0.1
7.9.2 HW0.0.3
James Thomas
10/18/1999