This course will introduces students to methods for solving partial differential equations (PDEs) using finite difference methods.
- Finite difference approximations of differential operators.
- Local and global truncation error; numerical consistency, stability and convergence
- The Fundamental Theorem of Finite Difference Methods.
- Steady state and boundary value problems
- Numerical time stepping methods for ordinary differential equations, including forward Euler, backward Euler, and multi-step and multi-stage (e.g. Runge-Kutta) methods.
- The time dependent heat equation (an example of a parabolic PDE), with particular focus on how to treat the stiffness inherent in parabolic PDEs. Method of lines discretizations. Lax-Equivalence Theorem; Lax-Richtmeyer Stability.
- Numerical methods for PDEs describing wave-like motion (hyperbolic PDEs).
- Numerical methods for mixed equations involving hyperbolic, parabolic and elliptic terms
- Brief introduction (if time permits) to finite volume methods for the discretization of conservation laws derived from physical principles.