Numerical Methods for Partial Differential Equations

1. Background Mathematics.
- 1.1 Introduction.
- 1.2 Vector and Matrix Norms.
- 1.3 Gerschgorin's Theorems.
- 1.4 Iterative Solution of Linear Algebraic Equations.
- 1.5 Further Results on Eigenvalues and Eigenvectors.
- 1.6 Classification of Second Order Partial Differential Equations.
- 2. Finite Differences and Parabolic Equations.
- 2.1 Finite Difference Approximations to Derivatives.
- 2.2 Parabolic Equations.
- 2.3 Local Truncation Error.
- 2.4 Consistency.
- 2.5 Convergence.
- 2.6 Stability.
- 2.7 The Crank-Nicolson Implicit Method.
- 2.8 Parabolic Equations in Cylindrical and Spherical Polar Coordinates.
- 3. Hyperbolic Equations and Characteristics.
- 3.1 First Order Quasi-linear Equations.
- 3.2 Lax-Wendroff and Wendroff Methods.
- 3.3 Second Order Quasi-linear Hyperbolic Equations.
- 3.4 Reetangular Nets and Finite Difference Methods for Second Order Hyperbolic Equations.
- 4. Elliptic Equations.
- 4.1 Laplace's Equation.
- 4.2 Curved Boundaries.
- 4.3 Solution of Sparse Systems of Linear Equations.
- 5. Finite Element Method for Ordinary Differential Equations.
- 5.1 Introduction.
- 5.2 The Collocation Method.
- 5.3 The Least Squares Method.
- 5.4 The Galerkin Method.
- 5.5 Symmetrie Variational Forrnulation.
- 5.6 Finite Element Method.
- 5.7 Some Worked Examples.
- 6. Finite Elements for Partial Differential Equations.
- 6.1 Introduction.
- 6.2 Variational Methods.
- 6.3 Some Specific Elements.
- 6.4 Assembly of the Elements.
- 6.5 Worked Example.
- 6.6 A General Variational Principle.
- 6.7 Assembly and Solution.
- 6.8 Solution of the Worked Example.
- 6.9 Further Interpolation Functions.
- 6.10 Quadrature Methods and Storage Considerations.
- 6.11 Boundary Element Method.- A. Solutions to Exercises.- References and Further Reading.