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Informationen zum Autor Dr. Ruas is currently a researcher in the Jean Le Rond d'Alembert Institute at the University ofPierre and Marie Curie. He was previously a Visiting Professor in mechanics and mathematics departments at the University of Tokyo, University of Hamburg and the University of São Paulo. His main areas of research cover Numerical Methods, Applied Mathematics and Fluid Flow Modeling. Klappentext Numerical Methods for Partial Differential Equations: An Introduction Vitoriano Ruas, Sorbonne Universités, UPMC - Université Paris 6, France A comprehensive overview of techniques for the computational solution of PDE's Numerical Methods for Partial Differential Equations: An Introduction covers the three most popular methods for solving partial differential equations: the finite difference method, the finite element method and the finite volume method. The book combines clear descriptions of the three methods, their reliability, and practical implementation aspects. Justifications for why numerical methods for the main classes of PDE's work or not, or how well they work, are supplied and exemplified. Aimed primarily at students of Engineering, Mathematics, Computer Science, Physics and Chemistry among others this book offers a substantial insight into the principles numerical methods in this class of problems are based upon. The book can also be used as a reference for research work on numerical methods for PDE's. Key features: - A balanced emphasis is given to both practical considerations and a rigorous mathematical treatment. - The reliability analyses for the three methods are carried out in a unified framework and in a structured and visible manner, for the basic types of PDE's. - Special attention is given to low order methods, as practitioner's overwhelming default options for everyday use. - New techniques are employed to derive known results, thereby simplifying their proof. - Supplementary material is available from a companion website. Inhaltsverzeichnis Preface by Eugenio Õnate xi Preface by Larisa Beilina xiii Acknowledgements xv About the Companion Website xvii Introduction xix Key Reminders on Linear Algebra xxvii 1 Getting Started in One Space Variable 1 1.1 A Model Two-point Boundary Value Problem 2 1.2 The Basic FDM 7 1.3 The Piecewise Linear FEM (P 1 FEM) 12 1.4 The Basic FVM 17 1.4.1 The Vertex-centred FVM 17 1.4.2 The Cell-centred FVM 20 1.4.3 Connections to the Other Methods 22 1.5 Handling Nonzero Boundary Conditions 24 1.6 Effective Resolution 25 1.6.1 Solving SLAEs for one-dimensional problems 26 1.6.2 Example 1.1: Numerical Experiments with the Cell-centred FVM 27 1.7 Exercises 28 2 Qualitative Reliability Analysis 30 2.1 Norms and Inner Products 31 2.1.1 Normed Vector Spaces 32 2.1.2 Inner Product Spaces 33 2.2 Stability of a Numerical Method 35 2.2.1 Stability in the Maximum Norm 35 2.2.2 Stability in the Mean-square Sense 39 2.3 Scheme Consistency 42 2.3.1 Consistency of the Three-point FD Scheme 42 2.3.2 Consistency of the P 1 FE Scheme 44 2.4 Convergence of the Discretisation Methods 48 2.4.1 Convergence of the Three-point FDM 49 2.4.2 Convergence of the P 1 FEM 50 2.4.3 Remarks on the Convergence of the FVM 52 2.4.4 Example 2.1: Sensitivity Study of Three Equivalent Methods 54 2.5 Exercises 59 3 Time-dependent Boundary Value Problems 61 3.1 Numerical Solution of the Heat Equation 64 3.1.1 Implicit Time Discretisation 65 3.1.2 Explicit Time Discretisation 66 3.1.3 Example 3.1: Numeric...
