MATH3081/4221 Numerical Differential Equations III/IV
Most differential equations, especially those of interest in real life, do not admit solutions in closed form. To study them, numerical methods are therefore essential. Applied correctly, numerical solutions become a powerful tool to tackle otherwise impossible problems, but when used incorrectly, one could be led to disastrously wrong conclusions (resulting in collapsing bridges, wayward spacecraft and bad weather forecast). Distinguishing correct and incorrect uses of numerical methods can be subtle and requires the understanding of both the numerics and the underlying problem. This is the main aim of this course.
In most the first term, we shall study various methods used to solve (meaning "obtain numerical approximate solutions of") ordinary differential equations, with emphasis on initial-value problems. Central to this is the notions of stability and convergence, which we will investigate mostly in the context of linear problems (with some nonlinear examples, which tend to be more difficult).
Following this, we shall look at numerical methods for partial differential equations. As with ODEs, the type of PDEs at hand (eg, parabolic, elliptic, hyperbolic, etc.) determines the methods that can be safely used, with the notions of stability and convergence again playing important roles.
This course consists of theory (eg, convergence proofs), explicit solutions (mostly for linear DEs) and computer exercises (both in class and for homework).
Outline of Course
Aim: To build a basic understanding of the theory, methods and practice of numerical solutions for differential equations, both ordinary and partial.
- Introduction to numerical methods for initial-value problems.
- Local and global truncation errors; convergence.
- One-step methods, with emphasis on explicit Runge-Kutta methods.
- Linear multistep methods, in particular the Adams methods.
- Predictor-corrector methods and estimation of local truncation error.
- Stability concepts and stiff problems.
- Finite difference methods for parabolic PDEs; CFL condition and stability.
- Finite-element method for elliptic equations.
- Iterative methods to solve linear systems.
- Eigenvalue problems for elliptic operators.
- Finite-volume method for the wave equations; CFL condition.