Course Description: Numerical methods for the solution of initial- and boundary-value problems for partial differential equations, with emphasis on finite difference methods. Consistency, stability, convergence, and implementation are considered.
Course Outline
- Review of classification of PDEs, Fourier analysis of PDEs, symbols of operators, dispersion relations, well posedness of initial-value and initial-boundary-value problems for model PDEs
- Numerical differentiation on a grid
- Some simple explicit & implicit Finite Difference (FD) numerical schemes for the model PDEs
- The concepts of Order of Accuracy, Stability, Consistency and Convergence of numerical schemes
- Fourier analysis on a grid. The Evaluation and Truncation operators.
- Stability of single- and multi-step FD schemes. Introduction to the effect of boundary conditions
- Dispersion and Dissipation of FD schemes
- Schemes for Hyperbolic and Parabolic PDEs and systems in 1 and 2 dimensions
- Stability analysis of Initial Boundary Value Problems for PDEs
- Applications to model linear and nonlinear (Newton iteration) PDEs