George F. Pinder, PhD, is a Distinguished Professor of Civil and Environmental Engineering with a secondary appointments in Mathematics and Statistics and Computer Science at the University of Vermont, Burlington, Vermont. He is the author or co-author of ten books in numerical mathematics and engineering. Dr. Pinder is the recipient of numerous national and international honors and is a member of the National Academy of Engineering.

• Preface vii1 Interpolation 11.1 Purpose 11.2 Definitions 11.3 Example 21.4 Weirstraus Approximation Theorem 31.5 Lagrange Interpolation 31.5.1 Example 61.6 Compare P2 (θ) and f (θ) 81.7 Error of Approximation 91.8 Multiple Elements 141.8.1 Example 171.9 Hermite Polynomials 191.10 Error in Approximation by Hermites 221.11 ChapterSummary 231.12 Problems 242 Numerical Differentiation 312.1 General Theory 312.2 Two-Point Difference Formulae 322.2.1 Forward Difference Formula 332.2.2 Backward Difference Formula 332.2.3 Example 342.2.4 Error of the Approximation 342.3 Two-Point Formulae from Taylor Series 362.4 Three-point Difference Formulae 382.4.1 First-Order Derivative Difference Formulae 392.4.2 Second-Order Derivatives 402.5 Chapter Summary 442.6 Problems 443 Numerical Integration 533.1 Newton-Cotes Quadrature Formula 533.1.1 Lagrange Interpolation 533.1.2 Trapezoidal Rule 543.1.3 Simpson’s Rule 553.1.4 General Form 563.1.5 Example using Simpson’s Rule 563.1.6 Gauss Legendre Quadrature 573.2 Chapter Summary 603.3 Problems 614 Initial Value Problems 654.1 Euler Forward Integration Method Example 664.2 Convergence 674.3 Consistency 704.4 Stability 714.4.1 Example of Stability 724.5 Lax Equivalence Theorem 724.6 Runge−Kutta Type Formulae 724.6.1 GeneralForm 724.6.2 Runge−Kutta First-Order Form (Euler’s Method) 734.6.3 Runge−Kutta Second-Order Form 734.7 ChapterSummary 764.8 Problems 765 Weighted Residuals Methods 815.1 Finite Volume or Subdomain Method 825.1.1 Example 845.1.2 Finite Difference Interpretation of the Finite Volume Method 915.2 Galerkin Method for First Order Equations 925.2.1 Finite-Difference Interpretation of the Galerkin Approximation 995.3 Galerkin Method for Second-Order Equations 995.3.1 Finite Difference Interpretation of Second-Order Galerkin Method 1075.4 Finite Volume Method for Second-Order Equations 1085.4.1 Example of Finite Volume Solution of a Second-Order Equation 1125.4.2 Finite Difference Representation of the Finite-Volume Method for Second-Order Equations 1185.5 CollocationMethod 1195.5.1 CollocationMethod forFirst-OrderEquations 1195.5.2 Collocation Method for Second-Order Equations 1225.6 ChapterSummary 1285.7 Problems 1286 Initial Boundary-Value Problems 1336.1 Introduction 1336.2 Two Dimensional Polynomial Approximations 1336.2.1 Example of a Two Dimensional Polynomial Approximation 1346.3 Finite Difference Approximation 1356.3.1 First-Order Accurate Finite Difference Calculation 1376.3.2 Example of Second Order Accurate Finite Difference Approximation in Space 1406.4 Stability of Finite Difference Approximations 1436.4.1 Example of Stability 1466.4.2 Example Simulation 1496.5 Galerkin Finite Element Approximations in Time 1516.5.1 Strategy One: Forward Difference Approximation 1536.5.2 Strategy Two: Backward Difference Approximation 1546.6 Chapter Summary 1556.7 Problems 1557 Finite Difference Methods in Two Space 1617.1 Example Problem 1667.2 Chapter Summary 1687.3 Problems 1688 Finite Element Methods in Two Space 1738.1 Finite Element Approximations over Rectangles 1738.2 Finite Element Approximations over Triangles 1868.2.1 Formulation of Triangular Basis Functions 1888.2.2 Example Problem of Finite Element Approximation over Triangles 1918.2.3 Second Type or Neumann Boundary-Value Problem 1988.3 Isoparametric Finite Element Approximation 2028.3.1 Natural Coordinate Systems 2028.3.2 Basis Functions 2088.3.3 Calculation of the Jacobian 2098.3.4 Example of Isoparametric Formulation 2138.4 Chapter Summary 2208.5 Problems 2209 Finite Volume Approximation in Two Space 2299.1 Finite Volume Formulation 2299.2 Finite Volume Example Problem 1 2359.2.1 Problem Definition 2359.2.2 Weighted Residual Formulation 2369.2.3 Element Coefficient Matrices 2379.2.4 Evaluation of the Line Integral 2389.2.5 Evaluation of the Area Integral 2459.2.6 Global Matrix Assembly 2499.3 Finite Volume Example Problem Two 2509.3.1 Problem Definition 2509.3.2 Weighted Residual Formulation 2519.3.3 Element Coefficient Matrices 2529.3.4 Evaluation of the Source Term 2539.4 Chapter Summary 2549.5 Problems 25410 Initial Boundary-Value Problems 26110.1 Mass Lumping 26310.2 Chapter Summary 26410.3 Problems 26411 Boundary-Value Problems in Three Space 26711.1 Finite Difference Approximations 26711.2 Finite Element Approximations 26811.3 Chapter Summary 27312 Nomenclature 277Index 281

George F. Pinder, PhD, is a Distinguished Professor of Civil and Environmental Engineering with a secondary appointments in Mathematics and Statistics and Computer Science at the University of Vermont, Burlington, Vermont. He is the author or co-author of ten books in numerical mathematics and engineering. Dr. Pinder is the recipient of numerous national and international honors and is a member of the National Academy of Engineering.

• Preface vii1 Interpolation 11.1 Purpose 11.2 Definitions 11.3 Example 21.4 Weirstraus Approximation Theorem 31.5 Lagrange Interpolation 31.5.1 Example 61.6 Compare P2 (θ) and f (θ) 81.7 Error of Approximation 91.8 Multiple Elements 141.8.1 Example 171.9 Hermite Polynomials 191.10 Error in Approximation by Hermites 221.11 ChapterSummary 231.12 Problems 242 Numerical Differentiation 312.1 General Theory 312.2 Two-Point Difference Formulae 322.2.1 Forward Difference Formula 332.2.2 Backward Difference Formula 332.2.3 Example 342.2.4 Error of the Approximation 342.3 Two-Point Formulae from Taylor Series 362.4 Three-point Difference Formulae 382.4.1 First-Order Derivative Difference Formulae 392.4.2 Second-Order Derivatives 402.5 Chapter Summary 442.6 Problems 443 Numerical Integration 533.1 Newton-Cotes Quadrature Formula 533.1.1 Lagrange Interpolation 533.1.2 Trapezoidal Rule 543.1.3 Simpson’s Rule 553.1.4 General Form 563.1.5 Example using Simpson’s Rule 563.1.6 Gauss Legendre Quadrature 573.2 Chapter Summary 603.3 Problems 614 Initial Value Problems 654.1 Euler Forward Integration Method Example 664.2 Convergence 674.3 Consistency 704.4 Stability 714.4.1 Example of Stability 724.5 Lax Equivalence Theorem 724.6 Runge−Kutta Type Formulae 724.6.1 GeneralForm 724.6.2 Runge−Kutta First-Order Form (Euler’s Method) 734.6.3 Runge−Kutta Second-Order Form 734.7 ChapterSummary 764.8 Problems 765 Weighted Residuals Methods 815.1 Finite Volume or Subdomain Method 825.1.1 Example 845.1.2 Finite Difference Interpretation of the Finite Volume Method 915.2 Galerkin Method for First Order Equations 925.2.1 Finite-Difference Interpretation of the Galerkin Approximation 995.3 Galerkin Method for Second-Order Equations 995.3.1 Finite Difference Interpretation of Second-Order Galerkin Method 1075.4 Finite Volume Method for Second-Order Equations 1085.4.1 Example of Finite Volume Solution of a Second-Order Equation 1125.4.2 Finite Difference Representation of the Finite-Volume Method for Second-Order Equations 1185.5 CollocationMethod 1195.5.1 CollocationMethod forFirst-OrderEquations 1195.5.2 Collocation Method for Second-Order Equations 1225.6 ChapterSummary 1285.7 Problems 1286 Initial Boundary-Value Problems 1336.1 Introduction 1336.2 Two Dimensional Polynomial Approximations 1336.2.1 Example of a Two Dimensional Polynomial Approximation 1346.3 Finite Difference Approximation 1356.3.1 First-Order Accurate Finite Difference Calculation 1376.3.2 Example of Second Order Accurate Finite Difference Approximation in Space 1406.4 Stability of Finite Difference Approximations 1436.4.1 Example of Stability 1466.4.2 Example Simulation 1496.5 Galerkin Finite Element Approximations in Time 1516.5.1 Strategy One: Forward Difference Approximation 1536.5.2 Strategy Two: Backward Difference Approximation 1546.6 Chapter Summary 1556.7 Problems 1557 Finite Difference Methods in Two Space 1617.1 Example Problem 1667.2 Chapter Summary 1687.3 Problems 1688 Finite Element Methods in Two Space 1738.1 Finite Element Approximations over Rectangles 1738.2 Finite Element Approximations over Triangles 1868.2.1 Formulation of Triangular Basis Functions 1888.2.2 Example Problem of Finite Element Approximation over Triangles 1918.2.3 Second Type or Neumann Boundary-Value Problem 1988.3 Isoparametric Finite Element Approximation 2028.3.1 Natural Coordinate Systems 2028.3.2 Basis Functions 2088.3.3 Calculation of the Jacobian 2098.3.4 Example of Isoparametric Formulation 2138.4 Chapter Summary 2208.5 Problems 2209 Finite Volume Approximation in Two Space 2299.1 Finite Volume Formulation 2299.2 Finite Volume Example Problem 1 2359.2.1 Problem Definition 2359.2.2 Weighted Residual Formulation 2369.2.3 Element Coefficient Matrices 2379.2.4 Evaluation of the Line Integral 2389.2.5 Evaluation of the Area Integral 2459.2.6 Global Matrix Assembly 2499.3 Finite Volume Example Problem Two 2509.3.1 Problem Definition 2509.3.2 Weighted Residual Formulation 2519.3.3 Element Coefficient Matrices 2529.3.4 Evaluation of the Source Term 2539.4 Chapter Summary 2549.5 Problems 25410 Initial Boundary-Value Problems 26110.1 Mass Lumping 26310.2 Chapter Summary 26410.3 Problems 26411 Boundary-Value Problems in Three Space 26711.1 Finite Difference Approximations 26711.2 Finite Element Approximations 26811.3 Chapter Summary 27312 Nomenclature 277Index 281

George F. Pinder, PhD, is a Distinguished Professor of Civil and Environmental Engineering with a secondary appointments in Mathematics and Statistics and Computer Science at the University of Vermont, Burlington, Vermont. He is the author or co-author of ten books in numerical mathematics and engineering. Dr. Pinder is the recipient of numerous national and international honors and is a member of the National Academy of Engineering.

• Preface vii1 Interpolation 11.1 Purpose 11.2 Definitions 11.3 Example 21.4 Weirstraus Approximation Theorem 31.5 Lagrange Interpolation 31.5.1 Example 61.6 Compare P2 (θ) and f (θ) 81.7 Error of Approximation 91.8 Multiple Elements 141.8.1 Example 171.9 Hermite Polynomials 191.10 Error in Approximation by Hermites 221.11 ChapterSummary 231.12 Problems 242 Numerical Differentiation 312.1 General Theory 312.2 Two-Point Difference Formulae 322.2.1 Forward Difference Formula 332.2.2 Backward Difference Formula 332.2.3 Example 342.2.4 Error of the Approximation 342.3 Two-Point Formulae from Taylor Series 362.4 Three-point Difference Formulae 382.4.1 First-Order Derivative Difference Formulae 392.4.2 Second-Order Derivatives 402.5 Chapter Summary 442.6 Problems 443 Numerical Integration 533.1 Newton-Cotes Quadrature Formula 533.1.1 Lagrange Interpolation 533.1.2 Trapezoidal Rule 543.1.3 Simpson’s Rule 553.1.4 General Form 563.1.5 Example using Simpson’s Rule 563.1.6 Gauss Legendre Quadrature 573.2 Chapter Summary 603.3 Problems 614 Initial Value Problems 654.1 Euler Forward Integration Method Example 664.2 Convergence 674.3 Consistency 704.4 Stability 714.4.1 Example of Stability 724.5 Lax Equivalence Theorem 724.6 Runge−Kutta Type Formulae 724.6.1 GeneralForm 724.6.2 Runge−Kutta First-Order Form (Euler’s Method) 734.6.3 Runge−Kutta Second-Order Form 734.7 ChapterSummary 764.8 Problems 765 Weighted Residuals Methods 815.1 Finite Volume or Subdomain Method 825.1.1 Example 845.1.2 Finite Difference Interpretation of the Finite Volume Method 915.2 Galerkin Method for First Order Equations 925.2.1 Finite-Difference Interpretation of the Galerkin Approximation 995.3 Galerkin Method for Second-Order Equations 995.3.1 Finite Difference Interpretation of Second-Order Galerkin Method 1075.4 Finite Volume Method for Second-Order Equations 1085.4.1 Example of Finite Volume Solution of a Second-Order Equation 1125.4.2 Finite Difference Representation of the Finite-Volume Method for Second-Order Equations 1185.5 CollocationMethod 1195.5.1 CollocationMethod forFirst-OrderEquations 1195.5.2 Collocation Method for Second-Order Equations 1225.6 ChapterSummary 1285.7 Problems 1286 Initial Boundary-Value Problems 1336.1 Introduction 1336.2 Two Dimensional Polynomial Approximations 1336.2.1 Example of a Two Dimensional Polynomial Approximation 1346.3 Finite Difference Approximation 1356.3.1 First-Order Accurate Finite Difference Calculation 1376.3.2 Example of Second Order Accurate Finite Difference Approximation in Space 1406.4 Stability of Finite Difference Approximations 1436.4.1 Example of Stability 1466.4.2 Example Simulation 1496.5 Galerkin Finite Element Approximations in Time 1516.5.1 Strategy One: Forward Difference Approximation 1536.5.2 Strategy Two: Backward Difference Approximation 1546.6 Chapter Summary 1556.7 Problems 1557 Finite Difference Methods in Two Space 1617.1 Example Problem 1667.2 Chapter Summary 1687.3 Problems 1688 Finite Element Methods in Two Space 1738.1 Finite Element Approximations over Rectangles 1738.2 Finite Element Approximations over Triangles 1868.2.1 Formulation of Triangular Basis Functions 1888.2.2 Example Problem of Finite Element Approximation over Triangles 1918.2.3 Second Type or Neumann Boundary-Value Problem 1988.3 Isoparametric Finite Element Approximation 2028.3.1 Natural Coordinate Systems 2028.3.2 Basis Functions 2088.3.3 Calculation of the Jacobian 2098.3.4 Example of Isoparametric Formulation 2138.4 Chapter Summary 2208.5 Problems 2209 Finite Volume Approximation in Two Space 2299.1 Finite Volume Formulation 2299.2 Finite Volume Example Problem 1 2359.2.1 Problem Definition 2359.2.2 Weighted Residual Formulation 2369.2.3 Element Coefficient Matrices 2379.2.4 Evaluation of the Line Integral 2389.2.5 Evaluation of the Area Integral 2459.2.6 Global Matrix Assembly 2499.3 Finite Volume Example Problem Two 2509.3.1 Problem Definition 2509.3.2 Weighted Residual Formulation 2519.3.3 Element Coefficient Matrices 2529.3.4 Evaluation of the Source Term 2539.4 Chapter Summary 2549.5 Problems 25410 Initial Boundary-Value Problems 26110.1 Mass Lumping 26310.2 Chapter Summary 26410.3 Problems 26411 Boundary-Value Problems in Three Space 26711.1 Finite Difference Approximations 26711.2 Finite Element Approximations 26811.3 Chapter Summary 27312 Nomenclature 277Index 281