Numerical Methods for Partial Differential Equations

Preface to Second EditionPreface to First Edition1 Fundamentals 1-0 Introduction 1-1 Classification of Physical Problems 1-2 Classification of Equations 1-3 Asymptotics 1-4 Discrete Methods 1-5 Finite Differences and Computational Molecules 1-6 Finite Difference Operators 1-7 Errors 1-8 Stability and Convergence 1-9 Irregular Boundaries 1-10 Choice of Discrete Network 1-11 Dimensionless Forms References2 Parabolic Equations 2-0 Introduction 2-1 Simple Explicit Methods 2-2 Fourier Stability Method 2-3 Implicit Methods 2-4 An Unconditionally Unstable Difference Equation 2-5 Matrix Stability Analysis 2-6 Extension of Matrix Stability Analysis 2-7 Consistency, Stability, and Convergence 2-8 Pure Initial Value Problems 2-9 Variable Coefficients 2-10 Examples of Equations with Variable Coefficients 2-11 General Concepts of Error Reduction 2-12 Explicit Methods for Nonlinear Problems 2-13 An Application of the Explicit Method 2-14 Implicit Methods for Nonlinear Problems 2-15 Concluding Remarks References3 Elliptic Equations 3-0 Introduction 3-1 Simple Finite Difference Schemes 3-2 Iterative Methods 3-3 Linear Elliptic Equations 3-4 Some Point Iterative Methods 3-5 Convergence of Point Iterative Methods 3-6 Rates of Convergence 3-7 Accelerations-Successive Over-Relaxation (SOR) 3-8 Extensions of SOR 3-9 Qualitative Examples of Over-Relaxation 3-10 Other Point Iterative Methods 3-11 Block Iterative Methods 3-12 Alternating Direction Methods 3-13 Summary of ADI Results 3-14 Some Nonlinear Examples References4 Hyperbolic Equations 4-0 Introduction 4-1 The Quasilinear System 4-2 Introductory Examples 4-3 Method of Characteristics 4-4 Constant States and Simple Waves 4-5 Typical Application of Characteristics 4-6 Explicit Finite Difference Methods 4-7 Overstability 4-8 Implicit Methods for Second-Order Equations 4-9 Nonlinear Examples 4-10 Simultaneous First-Order Equations-Explicit Methods 4-11 An Implicit Method for First-Order Equations 4-12 Hybrid Methods for First-Order Equations 4-13 Gas Dynamics in One-Space Variable 4-14 Eulerian Difference Equations 4-15 Lagrangian Difference Equations 4-16 Hopscotch Methods for Conservation Laws 4-17 Explicit-Implicit Schemes for Conservation Laws References5 Special Topics 5-0 Introduction 5-1 Singularities 5-2 Shocks 5-3 Eigenvalue Problems 5-4 Parabolic Equations in Several Space Variables 5-5 Additional Comments on Elliptic Equations 5-6 Hyperbolic Equations in Higher Dimensions 5-7 Mixed Systems 5-8 Higher-Order Equations in Elasticity and Vibrations 5-9 Fluid Mechanics: The Navier-Stokes Equations 5-10 Introduction to Monte Carlo Methods 5-11 Method of Lines 5-12 Fast Fourier Transform and Applications 5-13 Method of Fractional Steps References6 Weighted Residuals and Finite Elements 6-0 Introduction 6-1 Weighted Residual Methods (WRM) 6-2 Orthogonal Collocation 6-3 Bubnov-Galerkin (B-G) Method 6-4 Remarks on Completeness, Convergence, and Errors Bounds 6-5 Nagumo's Lemma and Application 6-6 Introduction to Finite Elements ReferencesAuthor IndexSubject Index