Mathematical Analysis and Numerical Methods for Science and Technology / Mathematical Analysis and Numerical Methods for Science and Technology Vol.6, Pt.2
Mitarbeit:Mercier, B.; Kavenoky, A.; Scheurer, B.; Sentis, R.; Pironneau, O.; Lascaux, P.; Bardos, C.; Cessenat, M.; Sneddon, I.N.;Übersetzung:Craig, A.
Mathematical Analysis and Numerical Methods for Science and Technology / Mathematical Analysis and Numerical Methods for Science and Technology Vol.6, Pt.2
Mitarbeit:Mercier, B.; Kavenoky, A.; Scheurer, B.; Sentis, R.; Pironneau, O.; Lascaux, P.; Bardos, C.; Cessenat, M.; Sneddon, I.N.;Übersetzung:Craig, A.
These six volumes - the result of a ten year collaboration between the authors, two of France's leading scientists and both 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. Since the publication in 1924 of the Methoden der mathematischen Physik by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in…mehr
These six volumes - the result of a ten year collaboration between the authors, two of France's leading scientists and both 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. Since the publication in 1924 of the Methoden der mathematischen Physik by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to caluclate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every fact of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences. Volumes 5 and 6 cover problems of Transport and Evolution.
Contributions by C. Bardos, M. Cessenat, A. Kavenoky, P. Lascaux, B. Mercier, O. Pironneau, B. Scheurer und R. Sentis
Inhaltsangabe
'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 Operato
'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 Operato
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