Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

https://hal.archives-ouvertes.fr/hal-01513346/document

Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome.

Spectral Methods in Fluid Dynamics

The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics In spite of the sophistication of the recently proposed methods, m

/pdf/a-review-of-image-denoising-algorithms-with-a-new-one-4vfxkcsntt.pdf

A Review of Image Denoising Algorithms, with a New One

I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Numerical methods for conservation laws

1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.2.1. A Fourier Galerkin Method for the Wave Equation.- 1.2.2. A Chebyshev Collocation Method for the Heat Equation.- 1.2.3. A Legendre Tau Method for the Poisson Equation.- 1.2.4. Basic Aspects of Galerkin, Tau and Collocation Methods.- 1.3. The Equations of Fluid Dynamics.- 1.3.1. Compressible Navier-Stokes.- 1.3.2. Compressible Euler.- 1.3.3. Compressible Potential.- 1.3.4. Incompressible Flow.- 1.3.5. Boundary Layer.- 1.4. Spectral Accuracy for a Two-Dimensional Fluid Calculation.- 1.5. Three-Dimensional Applications in Fluids.- 2. Spectral Approximation.- 2.1. The Fourier System.- 2.1.1. The Continuous Fourier Expansion.- 2.1.2. The Discrete Fourier Expansion.- 2.1.3. Differentiation.- 2.1.4. The Gibbs Phenomenon.- 2.2. Orthogonal Polynomials in ( - 1, 1).- 2.2.1. Sturm-Liouville Problems.- 2.2.2. Orthogonal Systems of Polynomials.- 2.2.3. Gauss-Type Quadratures and Discrete Polynomial Transforms.- 2.3. Legendre Polynomials.- 2.3.1. Basic Formulas.- 2.3.2. Differentiation.- 2.4. Chebyshev Polynomials.- 2.4.1. Basic Formulas.- 2.4.2. Differentiation.- 2.5. Generalizations.- 2.5.1. Jacobi Polynomials.- 2.5.2. Mapping.- 2.5.3. Semi-Infinite Intervals.- 2.5.4. Infinite Intervals.- 3. Fundamentals of Spectral Methods for PDEs.- 3.1. Spectral Projection of the Burgers Equation.- 3.1.1. Fourier Galerkin.- 3.1.2. Fourier Collocation.- 3.1.3. Chebyshev Tau.- 3.1.4. Chebyshev Collocation.- 3.2. Convolution Sums.- 3.2.1. Pseudospectral Transform Methods.- 3 2 2 Aliasing Removal by Padding or Truncation.- 3.2.3. Aliasing Removal by Phase Shifts.- 3.2.4. Convolution Sums in Chebyshev Methods.- 3.2.5. Relation Between Collocation and Pseudospectral Methods.- 3.3. Boundary Conditions.- 3.4. Coordinate Singularities.- 3.4.1. Polar Coordinates.- 3.4.2. Spherical Polar Coordinates.- 3.5. Two-Dimensional Mapping.- 4. Temporal Discretization.- 4.1. Introduction.- 4.2. The Eigenvalues of Basic Spectral Operators.- 4.2.1. The First-Derivative Operator.- 4.2.2. The Second-Derivative Operator.- 4.3. Some Standard Schemes.- 4.3.1. Multistep Schemes.- 4.3.2. Runge-Kutta Methods.- 4.4. Special Purpose Schemes.- 4.4.1. High Resolution Temporal Schemes.- 4.4.2. Special Integration Techniques.- 4.4.3. Lerat Schemes.- 4.5. Conservation Forms.- 4.6. Aliasing.- 5. Solution Techniques for Implicit Spectral Equations.- 5.1. Direct Methods.- 5.1.1. Fourier Approximations.- 5.1.2. Chebyshev Tau Approximations.- 5.1.3. Schur-Decomposition and Matrix-Diagonalization.- 5.2. Fundamentals of Iterative Methods.- 5.2.1. Richardson Iteration.- 5.2.2. Preconditioning.- 5.2.3. Non-Periodic Problems.- 5.2.4. Finite-Element Preconditioning.- 5.3. Conventional Iterative Methods.- 5.3.1. Descent Methods for Symmetric, Positive-Definite Systems.- 5.3.2. Descent Methods for Non-Symmetric Problems.- 5.3.3. Chebyshev Acceleration.- 5.4. Multidimensional Preconditioning.- 5.4.1. Finite-Difference Solvers.- 5.4.2. Modified Finite-Difference Preconditioners.- 5.5. Spectral Multigrid Methods.- 5.5.1. Model Problem Discussion.- 5.5.2. Two-Dimensional Problems.- 5.5.3. Interpolation Operators.- 5.5.4. Coarse-Grid Operators.- 5.5.5. Relaxation Schemes.- 5.6. A Semi-Implicit Method for the Navier-Stokes Equations.- 6. Simple Incompressible Flows.- 6.1. Burgers Equation.- 6.2. Shear Flow Past a Circle.- 6.3. Boundary-Layer Flows.- 6.4. Linear Stability.- 7. Some Algorithms for Unsteady Navier-Stokes Equations.- 7.1. Introduction.- 7.2. Homogeneous Flows.- 7.2.1. A Spectral Galerkin Solution Technique.- 7.2.2. Treatment of the Nonlinear Terms.- 7.2.3. Refinements.- 7.2.4. Pseudospectral and Collocation Methods.- 7.3. Inhomogeneous Flows.- 7.3.1. Coupled Methods.- 7.3.2. Splitting Methods.- 7.3.3. Galerkin Methods.- 7.3.4. Other Confined Flows.- 7.3.5. Unbounded Flows.- 7.3.6. Aliasing in Transition Calculations.- 7.4. Flows with Multiple Inhomogeneous Directions.- 7.4.1. Choice of Mesh.- 7.4.2. Coupled Methods.- 7.4.3. Splitting Methods.- 7.4.4. Other Methods.- 7.5. Mixed Spectral/Finite-Difference Methods.- 8. Compressible Flow.- 8.1. Introduction.- 8.2. Boundary Conditions for Hyperbolic Problems.- 8.3. Basic Results for Scalar Nonsmooth Problems.- 8.4. Homogeneous Turbulence.- 8.5. Shock-Capturing.- 8.5.1. Potential Flow.- 8.5.2. Ringleb Flow.- 8.5.3. Astrophysical Nozzle.- 8.6. Shock-Fitting.- 8.7. Reacting Flows.- 9. Global Approximation Results.- 9.1. Fourier Approximation.- 9.1.1. Inverse Inequalities for Trigonometric Polynomials.- 9.1.2. Estimates for the Truncation and Best Approximation Errors.- 9.1.3. Estimates for the Interpolation Error.- 9.2. Sturm-Liouville Expansions.- 9.2.1. Regular Sturm-Liouville Problems.- 9.2.2. Singular Sturm-Liouville Problems.- 9.3. Discrete Norms.- 9.4. Legendre Approximations.- 9.4.1. Inverse Inequalities for Algebraic Polynomials.- 9.4.2. Estimates for the Truncation and Best Approximation Errors.- 9.4.3. Estimates for the Interpolation Error.- 9.5. Chebyshev Approximations.- 9.5.1. Inverse Inequalities for Polynomials.- 9.5.2. Estimates for the Truncation and Best Approximation Errors.- 9.5.3. Estimates for the Interpolation Error.- 9.5.4. Proofs of Some Approximation Results.- 9.6. Other Polynomial Approximations.- 9.6.1. Jacobi Polynomials.- 9.6.2. Laguerre and Hermite Polynomials.- 9.7. Approximation Results in Several Dimensions.- 9.7.1. Fourier Approximations.- 9.7.2. Legendre Approximations.- 9.7.3. Chebyshev Approximations.- 9.7.4. Blended Fourier and Chebyshev Approximations.- 10. Theory of Stability and Convergence for Spectral Methods.- 10.1. The Three Examples Revisited.- 10.1.1. A Fourier Galerkin Method for the Wave Equation.- 10.1.2. A Chebyshev Collocation Method for the Heat Equation.- 10.1.3. A Legendre Tau Method for the Poisson Equation.- 10.2. Towards a General Theory.- 10.3. General Formulation of Spectral Approximations to Linear Steady Problems.- 10.4. Galerkin, Collocation and Tau Methods.- 10.4.1. Galerkin Methods.- 10.4.2. Tau Methods.- 10.4.3. Collocation Methods.- 10.5. General Formulation of Spectral Approximations to Linear Evolution Equations.- 10.5.1. Conditions for Stability and Convergence: The Parabolic Case.- 10.5.2. Conditions for Stability and Convergence: The Hyperbolic Case.- 10.6. The Error Equation.- 11. Steady, Smooth Problems.- 11.1. The Poisson Equation.- 11.1.1. Legendre Methods.- 11.1.2. Chebyshev Methods.- 11.1.3. Other Boundary Value Problems.- 11.2. Advection-Diffusion Equation.- 11.2.1. Linear Advection-Diffusion Equation.- 11.2.2. Steady Burgers Equation.- 11.3. Navier-Stokes Equations.- 11.3.1. Compatibility Conditions Between Velocity and Pressure.- 11.3.2. Direct Discretization of the Continuity Equation: The \"inf-sup\" Condition.- 11.3.3. Discretizations of the Continuity Equation by an Influence-Matrix Technique: The Kleiser-Schumann Method.- 11.3.4. Navier-Stokes Equations in Streamfunction Formulation.- 11.4. The Eigenvalues of Some Spectral Operators.- 11.4.1. The Discrete Eigenvalues for Lu = ? uxx.- 11.4.2. The Discrete Eigenvalues for Lu = ? vuxx + bux.- 11.4.3. The Discrete Eigenvalues for Lu = ux.- 12. Transient, Smooth Problems.- 12.1. Linear Hyperbolic Equations.- 12.1.1. Periodic Boundary Conditions.- 12.1.2. Non-Periodic Boundary Conditions.- 12.1.3. Hyperbolic Systems.- 12.1.4. Spectral Accuracy for Non-Smooth Solutions.- 12.2. Heat Equation.- 12.2.1. Semi-Discrete Approximation.- 12.2.2. Fully Discrete Approximation.- 12.3. Advection-Diffusion Equation.- 12.3.1. Semi-Discrete Approximation.- 12.3.2. Fully Discrete Approximation.- 13. Domain Decomposition Methods.- 13.1. Introduction.- 13.2. Patching Methods.- 13.2.1. Notation.- 13.2.2. Discretization.- 13.2.3. Solution Techniques.- 13.2.4. Examples.- 13.3. Variational Methods.- 13.3.1. Formulation.- 13.3.2. The Spectral-Element Method.- 13.4. The Alternating Schwarz Method.- 13.5. Mathematical Aspects of Domain Decomposition Methods.- 13.5.1. Patching Methods.- 13.5.2. Equivalence Between Patching and Variational Methods.- 13.6. Some Stability and Convergence Results.- 13.6.1. Patching Methods.- 13.6.2. Variational Methods.- Appendices.- A. Basic Mathematical Concepts.- B. Fast Fourier Transforms.- C. Jacobi-Gauss-Lobatto Roots.- References.

Spectral Methods in Fluid Dynamics.

In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

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Strong Stability-Preserving High-Order Time Discretization Methods

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

Many of the recently developed high-resolution schemes for hyperbolic conservation laws are based on upwind dierencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the eld-by-eld decomposition which is required in order to identify the \direction of the wind." Instead, we propose to use as a building block the more robust Lax-Friedrichs (LxF) solver. The main advantage is simplicity: no Riemann problems are solved and hence eld-by-eld decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. We compensate for it by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of the results obtained by such convenient central dierencing is comparable with those of the upwind schemes. c Academic Press, Inc.

Non-oscillatory Central Dierencing for Hyperbolic Conservation Laws

https://ntrs.nasa.gov/api/citations/19890001211/downloads/19890001211.pdf

Eitan Tadmor

Papers

Spectral Methods in Fluid Dynamics.

Strong Stability-Preserving High-Order Time Discretization Methods

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

Non-oscillatory Central Dierencing for Hyperbolic Conservation Laws

Non-oscillatory central differencing for hyperbolic conservation laws