Time Dependent Phase Space Filters: A Stable Absorbing Boundary Condition

Preface
Acknowledgment
Contents
1 Introduction
	1.1 The Open Boundary Problem
		1.1.1 Filtering Approach
		1.1.2 Method Accuracy
		1.1.3 The Advantages of the Method
		1.1.4 The Disadvantages of the Method
		1.1.5 Multiscale Extension
	1.2 Comparison to Other Methods
		1.2.1 Absorbing Boundary Conditions
			Exact Dirichlet-Neumann Maps for the Schrödinger Equation
			Paradifferential Strategy
		1.2.2 Absorbing Potentials and Perfectly Matched Layers
			Absorbing Potentials
			Perfectly Matched Layers
2 Definitions, Notations and A Brief Introduction to Frames
	2.1 Definitions and Notations
	2.2 A Brief Introduction to Frames
		2.2.1 Windowed Fourier Transform
		2.2.2 Localization of Phase Space
		2.2.3 Distinguished Sets of Framelets, Framelet Functionals
3 Windowed Fourier Transforms and Space Phase Numerics
	3.1 Basic Definitions and Properties
	3.2 Characterizing the Dual Window
	3.3 Localization of Phases and Spaces
	3.4 Phase Space Numerics
4 Description of Time Dependent Phase Space Filters
	4.1 An Introductory Example
	4.2 The TDPSF Algorithm
	4.3 Implementation of the Algorithm
	4.4 An Intuition of the Algorithm
	4.5 Possible Enhancements
	4.6 Slow Waves Multiscale Resolution
5 A More Practical Discussion: How to Choose the Parameters
6 The Behavior of Gaussian Framelets Under the Free Flow
	6.1 Error Estimates
	6.2 Location of Each Framelet
7 Assumptions and Accuracy Estimates
	7.1 Assumptions
	7.2 Discussions on the Assumptions
	7.3 The Algorithm
	7.4 Choices of the Parameters
	7.5 Accuracy Estimates
		7.5.1 Local (1 Step) Error
		7.5.2 Global Error Estimates
	7.6 Remarks
		7.6.1 Almost Optimality of the Estimates
		7.6.2 Difference with the Dirichlet-to-Neumann Approach
		7.6.3 The Ping Pong Phenomenon
		7.6.4 Bounds on kinf
8 Discussions on the Assumptions
	8.1 Stationary Potentials
	8.2 Assumption 1
	8.3 Assumption 2
	8.4 Assumption 3
	8.5 Assumption 4
	8.6 Assumption 5
	8.7 Assumption 6
9 Proof of Theorem 7.5.6
	9.1 Estimates of E(t), Q(t)
	9.2 Estimates of E ( t )
	9.3 Slow Waves
	9.4 Estimates of Q ( t )
10 Proof of Theorems 7.5.4 and 7.5.5
	10.1 Outgoing Waves
	10.2 Residual Waves
11 Numerical Experiments
	11.1 Case 1: T+R=E
	11.2 Case 2: T+R ≠E
	11.3 On Assumption 4
		11.3.1 A Coincidence
		11.3.2 Motivation in Constructing the TDPSF
References