The Courant - Friedrichs - Lewy (CFL) condition : 80 years after its discovery

This volume comprises a carefully selected collection of articles emerging from and pertinent to the 2010 CFL-80 conference in Rio de Janeiro, celebrating the 80^th^ anniversary of the Courant-Friedrichs-Lewy (CFL) condition. A major result in the field of numerical analysis, the CFL condition has influenced the research of many important mathematicians over the past eight decades, and this work is meant to take stock of its most important and current applications. __The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery__ will be of interest to practicing mathematicians, engineers, physicists, and graduate students who work with numerical methods. Cover......Page 1 The Courant-Friedrichs-Lewy (CFL) Condition......Page 3 Foreword......Page 6 Contents......Page 9 Stability of Difference Schemes......Page 11 1 Introduction......Page 18 2 A Dialogue......Page 19 3 Mathematical Intuition......Page 26 4 Pólya......Page 28 5 Mental Models......Page 30 6 Mental Models Subject to Social Control......Page 34 7 Dewey and Pragmatism......Page 36 References......Page 38 1 Introduction......Page 40 2 Equations of Resistive MHD......Page 42 3 RKDG Methods for Resistive MHD......Page 43 3.2 Three-Dimensional Basis Functions......Page 44 3.3 Time Stepping......Page 46 4 Three-Dimensional Arc Simulations......Page 47 4.1 Arc Generation......Page 49 4.2 Effects of External Magnetic Fields......Page 50 5 Conclusion......Page 51 References......Page 52 1 Introduction......Page 53 2 The Advection-Diffusion Equation......Page 54 3.1 Space-Time Notation......Page 55 3.2 Approximation Spaces......Page 56 3.3 Weak Formulation on Each Space-Time Element......Page 57 3.5 The Geometric Conservation Law......Page 58 3.6 Existence and Uniqueness of the Approximate Solution......Page 59 4 Numerical Results......Page 61 4.2 Steady-State Boundary Layer Problem......Page 62 4.3 A Rotating Gaussian Pulse on a Moving/Deforming Mesh......Page 66 References......Page 69 1 Introduction......Page 72 2 The Basic Estimate......Page 73 4 Finding the Fiber......Page 74 5 Moving Along the Fiber......Page 76 7 Some Examples......Page 77 References......Page 81 On the Quadratic Finite Element Approximation of 1D Waves: Propagation, Observation, Control, and Numerical Implementation......Page 82 1 Preliminaries on the Continuous Model and Problem Formulation......Page 83 2 Preliminaries on Numerical Controls Using P1 and P2 Finite Element Approximations......Page 85 3 Implementation of the Bi-grid Algorithm and Numerical Results......Page 94 References......Page 105 Space-Time Adaptive Multiresolution Techniques for Compressible Euler Equations......Page 107 1 Introduction......Page 108 2 The Numerical Schemes......Page 109 MR Scheme......Page 110 MR/CT Scheme......Page 112 3.1 Lax Test-Case in 1D......Page 113 3.2 2D Test-Case: Lax-Liu Configuration #6......Page 117 4 Conclusions......Page 121 References......Page 122 1 Introduction......Page 124 2.1 Hyperbolic Systems of Balance Laws......Page 125 2.2 Chapman-Engskog-Type Expansion......Page 126 2.3 The Role of a Mathematical Entropy......Page 128 3.1 Euler Equations with Friction Term......Page 131 3.2 M1 Model of Radiative Transfer......Page 132 3.3 Coupled Euler/M1 Model......Page 133 3.4 Shallow Water with Nonlinear Friction......Page 134 4.1 General Strategy......Page 136 4.2 Discretization of the Relaxation Term......Page 138 4.3 Discrete Late-Time Asymptotic Regime......Page 139 References......Page 141 1 Introduction......Page 143 2.1 Runge-Kutta Schemes......Page 144 2.3 Stability......Page 145 2.4 A Refined CFL Condition and Stability......Page 146 3.1 Inviscid Burgers Equation......Page 147 3.2 Incompressible Euler and Navier-Stokes Equations......Page 148 4 Conclusions......Page 149 References......Page 150 1 Introduction......Page 151 2 Faster Gradient Descent Methods......Page 154 3 Chaos......Page 156 References......Page 158 Played by Lori Courant Lax, viola, Dorothy Strahl, violin, and Carol Buck, cello......Page 160 Appendix B......Page 163 Appendix C......Page 207 Technical sessions schedule......Page 228 List of supporting institutions......Page 232 List of lecturers, attendees, and organizers......Page 233 This volume comprises a carefully selected collection of articles emerging from and pertinent to the 2010 CFL-80 conference in Rio de Janeiro, celebrating the 80 th anniversary of the Courant–Friedrichs–Lewy (CFL) condition. A major result in the field of numerical analysis, the CFL condition has influenced the research of many important mathematicians over the past eight decades, and this work is meant to take stock of its most important and current applications. The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery will be of interest to practicing mathematicians, engineers, physicists, and graduate students who work with numerical methods. Contributors: U. Ascher B. Cockburn E. Deriaz M.O. Domingues S.M. Gomes R. Hersh R. Jeltsch D. Kolomenskiy H. Kumar L.C. Lax P. Lax P. LeFloch A. Marica O. Roussel K. Schneider J. Tiexeira Cal Neto C. Tomei K. van den Doel E. Zuazua This volume comprises a carefully selected collection of articles emerging from and pertinent to the 2010 CFL-80 conference in Rio de Janeiro, celebrating the 80th anniversary of the Courant-Friedrichs-Lewy (CFL) condition. A major result in the field of numerical analysis, the CFL condition has influenced the research of many important mathematicians over the past eight decades, and this work is meant to take stock of its most important and current applications. The Courant-Friedrichs-Lewy (CFL) Condition: 80 Years After its Discovery will be of interest to practicing mathematicians, engineers, physicists, and graduate students who work with numerical methods. Contributors: U. Ascher B. CockburnE. Deriaz M.O. Domingues S.M. GomesR. HershR. JeltschD. Kolomenskiy H. KumarL. C. Lax P. LaxP. LeFloch A. MaricaO. RousselK. SchneiderJ. Tiexeira Cal Neto C. TomeiK. van den DoelE. Zuazua Foreword.- Stability of Different Schemes.- Mathematical Intuition: Poincare, Polya, Dewey.- Three-dimensional Plasma Arc Simulation using Resistive MHD.- A Numerical Algorithm for Ambrosetti-Prodi Type Operators.- On the Quadratic Finite Element Approximation of 1-D Waves: Propagation, Observation, Control, and Numerical Implementation.- Space-Time Adaptive Mutilresolution Techniques for Compressible Euler Equations.- A Framework for Late-time/stiff Relaxation Asymptotics.- Is the CFL Condition Sufficient? Some Remarks.- Fast Chaotic Artificial Time Integration.- Appendix A.- Hans Lewy's Recovered String Trio.- Appendix B.- Appendix C.- Appendix D

Foreword Stability of Different Schemes Mathematical Intuition: Poincaré, Pólya, Dewey.- Three-dimensional Plasma Arc Simulation using Resistive MHD A Numerical Algorithm for Ambrosetti-Prodi Type Operators On the Quadratic Finite Element Approximation of 1-D Waves: Propagation, Observation, Control, and Numerical Implementation Space-Time Adaptive Mutilresolution Techniques for Compressible Euler Equations A Framework for Late-time/stiff Relaxation Asymptotics Is the CFL Condition Sufficient? Some Remarks Fast Chaotic Artificial Time Integration Appendix A Hans Lewy's Recovered String Trio Appendix B Appendix C Appendix D.

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