These six volumes--the result of a ten year collaboration between two distinguished international figures--compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. It is a comprehensive and up-to-date publication that presents the mathematical tools needed in applications of mathematics.
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.
