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Table of Contents
Intro
1 Introduction
2 Hyperbolic conservation laws
2.1 General form and examples
2.2 Classical solutions
2.3 Breakdown of classical solutions
2.4 Weak solutions
2.5 The Riemann problem
2.6 The role of viscosity
2.7 Entropy solutions
2.8 Some results for scalar conservation laws
3 Numerical preliminaries
3.1 Approximation and interpolation
3.1.1 Approximation in normed linear spaces
3.1.2 Approximation in inner product spaces
3.1.3 Polynomial interpolation in one dimension
3.1.4 General interpolation and the Mairhuber-Curtis theorem
3.1.5 Radial basis function interpolation
3.2 Orthogonal polynomials
3.2.1 Continuous and discrete inner products
3.2.2 Bases of orthogonal polynomials
3.2.3 Application to least squares approximations
3.3 Numerical differentiation
3.3.1 Finite difference approximations
3.3.2 Beyond finite differences
3.4 Numerical integration
3.4.1 The basic idea: Quadrature rules
3.4.2 What do we want? Exactness and stability
3.4.3 Interpolatory quadrature rules
3.5 Time integration
3.5.1 The method of lines
3.5.2 Preferred method for time integration
3.5.3 Strong stability of explicit Runge-Kutta methods
4 Stable high order quadrature rules for experimental data I: Nonnegative weight functions
4.1 Motivation
4.2 Least squares quadrature rules
4.2.1 Formulation as a least squares problem
4.2.2 Characterization by discrete orthonormal polynomials
4.2.3 Weighted least squares quadrature rules
4.3 Stability of least squares quadrature rules
4.3.1 Main result and consequences
4.3.2 Proof of the main result
4.4 Numerical tests
4.4.1 Comparison of different inner products
4.4.2 Minimal number of quadrature points for different weight functions
4.4.3 Accuracy on equidistant points.
4.4.4 Accuracy on scattered points
4.5 Concluding thoughts and outlook
5 Stable high order quadrature rules for experimental data II: General weight functions
5.1 Motivation
5.2 Stability concepts for general weight functions
5.3 Stability of least squares quadrature rules
5.3.1 Main results
5.3.2 Preliminaries on discrete Chebyshev polynomials
5.3.3 Proofs of the main results
5.4 Nonnegative least squares quadrature rules
5.4.1 The nonnegative least squares problem
5.4.2 Formulation as a nonnegative least squares problem
5.5 Numerical results
5.5.1 Implementation details
5.5.2 Stability
5.5.3 Sign-consistency
5.5.4 Exactness
5.5.5 Accuracy for increasing N
5.5.6 Accuracy for increasing d
5.5.7 Ratio between d and N
5.6 Concluding thoughts and outlook
6 High order numerical methods for conservation laws
6.1 Discontinuous Galerkin spectral element methods
6.1.1 Introduction
6.1.2 The analytical discontinuous Galerkin method
6.1.3 The discontinuous Galerkin collocation spectral element method
6.1.4 Matrix vector notation
6.2 Flux reconstruction methods
6.2.1 General idea
6.2.2 Flux reconstruction and summation by parts operators
6.2.3 Outlook on new stability results
7 Two novel high order methods
7.1 Stable discretizations of DG methods on equidistant and scattered points
7.1.1 Motivation
7.1.2 Discrete least squares approximations
7.1.3 Proposed discretization of the discontinuous Galerkin method
7.1.4 Conservation and stability
7.1.5 Numerical results
7.1.6 Concluding thoughts and outlook
7.2 Stable radial basis function methods
7.2.1 Motivation
7.2.2 State of the art: Stability of radial basis function methods
7.2.3 General idea
7.2.4 Weak radial basis function analytical methods.
7.2.5 Weak radial basis function collocation methods
7.2.6 Relationship to other methods
7.2.7 Efficient implementation
7.2.8 Numerical results
7.2.9 Concluding thoughts and outlook
8 Artificial viscosity methods
8.1 The idea behind artificial viscosity
8.2 State of the art
8.2.1 Persson and Peraire: Piecewise constant artificial viscosities
8.2.2 Barter and Darmofal: Smoothing the artificial viscosity
8.2.3 Klöckner et al.: Piecewise linear artificial viscosities
8.3 Conservation and stability properties
8.3.1 Conservation
8.3.2 Entropy dissipation
8.4 New viscosity distributions
8.5 Modal filtering
8.6 Discretization of artificial viscosity terms using SBP operators
8.7 Concluding thoughts
9 `1 regularization and high order edge sensors for enhanced discontinuous Galerkin methods
9.1 Why `1 regularization?
9.2 Preliminaries
9.2.1 `1 regularization
9.2.2 Polynomial annihilation
9.3 Application of `1 regularization to discontinuous Galerkin methods
9.3.1 Procedure
9.3.2 Selection of the regularization parameter
9.3.3 Discontinuity sensor
9.3.4 Efficient implementation of the PA operator
9.3.5 The alternating direction method of multipliers
9.3.6 Preservation of mass conservation
9.4 Numerical results
9.4.1 Inviscid Burgers' equation
9.4.2 Linear advection equation
9.4.3 Systems of nonlinear conservation laws
9.5 Concluding thoughts and outlook
10 Shock capturing by Bernstein polynomials
10.1 Bernstein polynomials and the Bernstein operator
10.1.1 Structure-preserving properties
10.1.2 Approximation properties
10.2 The Bernstein procedure
10.2.1 Related works
10.2.2 Bernstein reconstruction
10.2.3 Proposed procedure
10.2.4 Selection of parameter
10.2.5 Discontinuity sensor.
10.3 Entropy, total variation, and monotone (shock) profiles
10.3.1 Entropy stability
10.3.2 Total variation
10.3.3 Monotone (shock) profiles
10.4 Numerical results
10.4.1 Linear advection equation
10.4.2 Inviscid Burgers' equation
10.4.3 A concave flux function
10.4.4 The Buckley-Leverett equation
10.5 Concluding thoughts and outlook
11 High order edge sensor steered artificial viscosity operators
11.1 Generalized artificial viscosity operators
11.1.1 State of the art
11.1.2 Our contribution
11.2 Conservation and dissipation - continuous setting
11.2.1 Conservation
11.2.2 Energy dissipation
11.2.3 Entropy dissipation
11.2.4 In a nutshell
11.3 High order FD methods based on SBP operators
11.4 Conservation and dissipation - discrete setting
11.4.1 Conservation
11.4.2 Energy dissipation
11.4.3 Entropy dissipation
11.4.4 In a nutshell
11.5 High order edge sensor steered artificial viscsosity operators
11.5.1 High order edge sensors and polynomial annihilation
11.5.2 Distributing the viscosity
11.5.3 Scaling the viscosity strength
11.5.4 Choosing a sensing variable
11.6 Numerical results
11.6.1 Linear advection equation
11.6.2 Burgers' equation
11.6.3 Euler equations of gas dynamics
11.7 Concluding thoughts and outlook
12 Summary and outlook
Bibliography
Index
Glossary.
1 Introduction
2 Hyperbolic conservation laws
2.1 General form and examples
2.2 Classical solutions
2.3 Breakdown of classical solutions
2.4 Weak solutions
2.5 The Riemann problem
2.6 The role of viscosity
2.7 Entropy solutions
2.8 Some results for scalar conservation laws
3 Numerical preliminaries
3.1 Approximation and interpolation
3.1.1 Approximation in normed linear spaces
3.1.2 Approximation in inner product spaces
3.1.3 Polynomial interpolation in one dimension
3.1.4 General interpolation and the Mairhuber-Curtis theorem
3.1.5 Radial basis function interpolation
3.2 Orthogonal polynomials
3.2.1 Continuous and discrete inner products
3.2.2 Bases of orthogonal polynomials
3.2.3 Application to least squares approximations
3.3 Numerical differentiation
3.3.1 Finite difference approximations
3.3.2 Beyond finite differences
3.4 Numerical integration
3.4.1 The basic idea: Quadrature rules
3.4.2 What do we want? Exactness and stability
3.4.3 Interpolatory quadrature rules
3.5 Time integration
3.5.1 The method of lines
3.5.2 Preferred method for time integration
3.5.3 Strong stability of explicit Runge-Kutta methods
4 Stable high order quadrature rules for experimental data I: Nonnegative weight functions
4.1 Motivation
4.2 Least squares quadrature rules
4.2.1 Formulation as a least squares problem
4.2.2 Characterization by discrete orthonormal polynomials
4.2.3 Weighted least squares quadrature rules
4.3 Stability of least squares quadrature rules
4.3.1 Main result and consequences
4.3.2 Proof of the main result
4.4 Numerical tests
4.4.1 Comparison of different inner products
4.4.2 Minimal number of quadrature points for different weight functions
4.4.3 Accuracy on equidistant points.
4.4.4 Accuracy on scattered points
4.5 Concluding thoughts and outlook
5 Stable high order quadrature rules for experimental data II: General weight functions
5.1 Motivation
5.2 Stability concepts for general weight functions
5.3 Stability of least squares quadrature rules
5.3.1 Main results
5.3.2 Preliminaries on discrete Chebyshev polynomials
5.3.3 Proofs of the main results
5.4 Nonnegative least squares quadrature rules
5.4.1 The nonnegative least squares problem
5.4.2 Formulation as a nonnegative least squares problem
5.5 Numerical results
5.5.1 Implementation details
5.5.2 Stability
5.5.3 Sign-consistency
5.5.4 Exactness
5.5.5 Accuracy for increasing N
5.5.6 Accuracy for increasing d
5.5.7 Ratio between d and N
5.6 Concluding thoughts and outlook
6 High order numerical methods for conservation laws
6.1 Discontinuous Galerkin spectral element methods
6.1.1 Introduction
6.1.2 The analytical discontinuous Galerkin method
6.1.3 The discontinuous Galerkin collocation spectral element method
6.1.4 Matrix vector notation
6.2 Flux reconstruction methods
6.2.1 General idea
6.2.2 Flux reconstruction and summation by parts operators
6.2.3 Outlook on new stability results
7 Two novel high order methods
7.1 Stable discretizations of DG methods on equidistant and scattered points
7.1.1 Motivation
7.1.2 Discrete least squares approximations
7.1.3 Proposed discretization of the discontinuous Galerkin method
7.1.4 Conservation and stability
7.1.5 Numerical results
7.1.6 Concluding thoughts and outlook
7.2 Stable radial basis function methods
7.2.1 Motivation
7.2.2 State of the art: Stability of radial basis function methods
7.2.3 General idea
7.2.4 Weak radial basis function analytical methods.
7.2.5 Weak radial basis function collocation methods
7.2.6 Relationship to other methods
7.2.7 Efficient implementation
7.2.8 Numerical results
7.2.9 Concluding thoughts and outlook
8 Artificial viscosity methods
8.1 The idea behind artificial viscosity
8.2 State of the art
8.2.1 Persson and Peraire: Piecewise constant artificial viscosities
8.2.2 Barter and Darmofal: Smoothing the artificial viscosity
8.2.3 Klöckner et al.: Piecewise linear artificial viscosities
8.3 Conservation and stability properties
8.3.1 Conservation
8.3.2 Entropy dissipation
8.4 New viscosity distributions
8.5 Modal filtering
8.6 Discretization of artificial viscosity terms using SBP operators
8.7 Concluding thoughts
9 `1 regularization and high order edge sensors for enhanced discontinuous Galerkin methods
9.1 Why `1 regularization?
9.2 Preliminaries
9.2.1 `1 regularization
9.2.2 Polynomial annihilation
9.3 Application of `1 regularization to discontinuous Galerkin methods
9.3.1 Procedure
9.3.2 Selection of the regularization parameter
9.3.3 Discontinuity sensor
9.3.4 Efficient implementation of the PA operator
9.3.5 The alternating direction method of multipliers
9.3.6 Preservation of mass conservation
9.4 Numerical results
9.4.1 Inviscid Burgers' equation
9.4.2 Linear advection equation
9.4.3 Systems of nonlinear conservation laws
9.5 Concluding thoughts and outlook
10 Shock capturing by Bernstein polynomials
10.1 Bernstein polynomials and the Bernstein operator
10.1.1 Structure-preserving properties
10.1.2 Approximation properties
10.2 The Bernstein procedure
10.2.1 Related works
10.2.2 Bernstein reconstruction
10.2.3 Proposed procedure
10.2.4 Selection of parameter
10.2.5 Discontinuity sensor.
10.3 Entropy, total variation, and monotone (shock) profiles
10.3.1 Entropy stability
10.3.2 Total variation
10.3.3 Monotone (shock) profiles
10.4 Numerical results
10.4.1 Linear advection equation
10.4.2 Inviscid Burgers' equation
10.4.3 A concave flux function
10.4.4 The Buckley-Leverett equation
10.5 Concluding thoughts and outlook
11 High order edge sensor steered artificial viscosity operators
11.1 Generalized artificial viscosity operators
11.1.1 State of the art
11.1.2 Our contribution
11.2 Conservation and dissipation - continuous setting
11.2.1 Conservation
11.2.2 Energy dissipation
11.2.3 Entropy dissipation
11.2.4 In a nutshell
11.3 High order FD methods based on SBP operators
11.4 Conservation and dissipation - discrete setting
11.4.1 Conservation
11.4.2 Energy dissipation
11.4.3 Entropy dissipation
11.4.4 In a nutshell
11.5 High order edge sensor steered artificial viscsosity operators
11.5.1 High order edge sensors and polynomial annihilation
11.5.2 Distributing the viscosity
11.5.3 Scaling the viscosity strength
11.5.4 Choosing a sensing variable
11.6 Numerical results
11.6.1 Linear advection equation
11.6.2 Burgers' equation
11.6.3 Euler equations of gas dynamics
11.7 Concluding thoughts and outlook
12 Summary and outlook
Bibliography
Index
Glossary.