Stability in differential equations concerns the global effects of local perturbations. Many students of differential equaltions first learn about stability in the form of well-posedness and the classic Lax equivalence theorem (that well-posedness plus consistency equals convergence). However, many other notions of stability are equally important in practice, and this volume tackles the challenging concepts of stability beyond well-posedness. The lectures in this volume were chosen to strike a reasonable balance between dynamical and classical analysis, between structure-preserving and character-preserving numerics, and between the preservation of stability under discretization and the study of stability by computation. The broad range of topics presented in this book exposes many parallel themes. Armed with an understanding of the broader picture and in possession of a good set of references, the reader should then be prepared to seek a deeper comprehension of stability.
Table of Contents
List of Contributors; Preface; Chapter 1: Introduction, D. Estep and S. Tavener; Part I: Preservation of Qualitative Stability Features. Chapter 2: Preservation of Strong Stability Associated with Analytic Semigroups, S. Larsson; Chapter 3: From Semidiscrete to Fully Discrete: Stability of Runge-Kutta Schemes by the Energy Method. II., E. Tadmor; Chapter 4: A Survey of Strong Stability Preserving High Order Time Discretizations, C. W. Shu; Chapter 5: Continuous Dependence Results for Hamilton-Jacobi Equations, B. Cockburn and J. Qian; Part II: Preservation of Structural Stability. Chapter 6: The inf-sup Condition in Mixed Finite Element Methods with Application to the Stokes System, M. D. Gunzburger; Chapter 7: Brief Introduction to Lie-group Methods, A. Iserles; Chapter 8: Matching Algorithms with Physics: Exact Sequences of Finite Element Spaces, P. B. Bochev and A. C. Robinson; Part III: Investigation of Physical Stability. Chapter 9: Eigenvalue and Parameter Approximation at Hopf Bifurcation, A. Spence; Chapter 10: RPM-A Remedy for Instability, H. B. Keller; Chapter 11: Lyapunov and Other Spectra: A Survey, L. Dieci and E. S. Van Vleck; Part IV: Investigation of Model Stability. Chapter 12: Stability of Dispersive Model Equations for Fluid Flow, V. Girault and L. R. Scott; Chapter 13: Chaotic Dynamics in Nonlinear Waves: Computational and Physical, M. J. Ablowitz and C. M. Schober; Index.