Mathematical Analysis and Numerical Methods for Science and Technology
Volume 6 Evolution Problems II
Published on 23. November 1999
Book
Paperback/Softback
XII, 486 pages
978-3-540-66102-3 (ISBN)
Description
The object ofthis chapter is to present a certain number ofresults on the linearised Navier-Stokes equations. The Navier-Stokes equations, which describe the motion of a viscous, incompressible fluid were introduced already, from the physical point of view, in §1 of Chap. IA. These equations are nonlinear. We study here the equations that emerge on linearisation from the solution (u = 0, p = 0). This is an interesting exercise in its own right. It corresponds to the case of a very slow flow, and also prepares the way for the study of the complete Navier-Stokes equations. This Chap. XIX is made up of two parts, devoted respectively to linearised stationary equations (or Stokes' problem), and to linearised evolution equations. Questions of existence, uniqueness, and regularity of solutions are considered from the variational point of view, making use of general results proved elsewhere. The functional spaces introduced for this purpose are themselves of interest and are therefore studied comprehensively.
More details
Edition
2000
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Edition type
New edition
Illustrations
XII, 486 p.
Dimensions
Height: 23.5 cm
Width: 15.5 cm
Weight
1540 gr
ISBN-13
978-3-540-66102-3 (9783540661023)
DOI
10.1007/978-3-642-58004-8
Schweitzer Classification
Persons
Content
XIX. The Linearised Navier-Stokes Equations.- §1. The Stationary Navier-Stokes Equations: The Linear Case.- 1. Functional Spaces.- 2. Existence and Uniqueness Theorem.- 3. The Problem of L? Regularity.- §2. The Evolutionary Navier-Stokes Equations: The Linear Case.- 1. Functional Spaces and Trace Theorems.- 2. Existence and Uniqueness Theorem.- 3. L2-Regularity Result.- §3. Additional Results and Review.- 1. The Variational Approach.- 2. The Functional Approach.- 3. The Problem of L? Regularity for the Evolutionary Navier-Stokes Equations: The Linearised Case.- XX. Numerical Methods for Evolution Problems.- §1. General Points.- 1. Discretisation in Space and Time.- 2. Convergence, Consistency and Stability.- 3. Equivalence Theorem.- 4. Comments.- 5. Schemes with Constant Coefficients and Step Size.- 6. The Symbol of a Difference Scheme.- 7. The von Neumann Stability Condition.- 8. The Kreiss Stability Condition.- 9. The Case of Multilevel Schemes.- 10. Characterisation of a Scheme of Order q.- §2. Problems of First Order in Time.- 1. Introduction.- 2. Model Equation for x ? ?.- 3. The Boundary Value Problem for Equation.- 4. Equation with Variable Coefficients and Schemes with Variable Step-Size.- 5. The Heat Flow Equation in Two Space Dimensions.- 6. Alternating Direction and Fractional Step Methods.- 7. Internal Approximation Schemes.- 8. Integration of Systems of Stiff Differential Equations.- 9. Comments.- §3. Problems of Second Order in Time.- 1. Introduction.- 2. The Model Equation for x ? ?.- 3. The Wave Equation in Two Space Dimensions.- 4. Internal Approximation Schemes.- 5. The Newmark Scheme.- 6. The Wave Equation with Viscosity.- 7. The Wave Equation Coupled to a Heat Flow Equation.- 8. Comments.- §4. The Advection Equation.- 1. Introduction.- 2. Some Explicit Schemes for the Cauchy Problem in One Space Dimension.- 3. Positive-Type Schemes and Stable Schemes in LX(?).- 4. Some Explicit Schemes.- 5. The Problem with Boundary Conditions.- 6. Phase and Amplitude Error. Schemes of Order Greater than Two.- 7. Nonlinear Schemes for the Equation.- 8. Difference Schemes for the Cauchy Problem with Many Space Variables.- §5. Symmetric Friedrichs Systems.- 1. Introduction.- 2. Summary of Symmetric Friedrichs Systems.- 3. Finite Difference Schemes for the Cauchy Problem.- 4. Approximation of Boundary Conditions in the Case where ? = ]0, 1 [.- 5. Maxwell's Equations.- 6. Remarks.- §6. The Transport Equation.- 1. Introduction.- 2. Stationary Equation in One-Dimensional Plane Geometry.- 3. The Evolution Equation in One-Dimensional Plane Geometry.- 4. The Equation in One-Dimensional Spherical Geometry.- 5. Iterative Solution of Schemes Approximating the Transport Equation.- 6. The Two-Dimensional Equation.- 7. Other Methods.- 8. Comments.- §7. Numerical Solution of the Stokes Problem.- 1. Setting of Problem.- 2. An Integral Method.- 3. Some Finite Difference Methods.- 4. Finite Element Methods.- 5. Some Methods Using the Stream function.- 6. The Evolutionary Stokes Problem.- XXI. Transport.- §1. Introduction. Presentation of Physical Problems.- 1. Evolution Problems in Neutron Transport.- 2. Stationary Problems.- 3. Principal Notation.- §2. Existence and Uniqueness of Solutions of the Transport Equation.- 1. Introduction.- 2. Study of the Advection Operator A = - v. ?.- 3. Solution of the Cauchy Transport Problem.- 4. Solution of the Stationary Transport Problem in the Subcritical Case.- Summary.- Appendix of §2. Boundary Conditions in Transport Problems. Reflection Conditions.- §3. Spectral Theory and Asymptotic Behaviour of the Solutions of Evolution Problems.- 1. Introduction.- 2. Study of the Spectrum of the Operator B = - v. ? - ?.- 3. Study of the Spectrum of the Transport Operator in an Open Bounded Set X of ?n.- 4. Positivity Properties.- 5. The Particular Case where All the Eigenvalues are Real.- 6. The Spectrum of the Transport Operator in a Band. The Lehner-Wing Theorem.- 7. Study of the Spectrum of the Transport Operator in the Whole Space: X = ?n.- 8. The Spectrum of the Transport Operator on the Exterior of an "Obstacle".- 9. Some Remarks on the Spectrum of T.- Summary.- Appendix of §3. The Conservative Milne Problem.- §4. Explicit Examples.- 1. The Stationary Transport Problem in the Whole Space ?.- 2. The Evolutionary Transport Problem in the Whole Space.- 3. The Stationary Transport Problem in the Half-Space by the Method of "Invariant Embedding".- 4. Case's Method of "Generalised Eigenfunctions". Application to the Critical Dimension in the Case of a Band.- §5. Approximation of the Neutron Transport Equation by the Diffusion Equation.- 1. Physical Introduction.- 2. Approximation in the Case of a Monokinetic Model of Evolution Equations and of Stationary Transport Equations.- 3. Generalisation of Section 2.- 4. Calculation of a Corrector for the Stationary Problem and Extrapolation Length.- 5. Convergence of the Principal Eigenvalue of the Transport Operator.- 6. Calculation of a Corrector for the Principal Eigenvalue of the Transport Operator.- 7. Application to a Critical Size Problem.- 8. Numerical Example in the Case of a Band.- Appendix of §5.- Perspectives.- Orientation for the Reader.- List of Equations.- Table of Notations.- Cumulative Index of Volumes 1-6.- of Volumes 1-5.