Mathematical Analysis and Numerical Methods for Science and Technology: Evolution Problems II: Vol 6

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.