Numerical methods 4010-MN-30

1. Introduction to numerical solutions of differential equations. Simple schemes: linear, single-step, open, and closed. Taylor-type methods. Open and closed Runge-Kutta-type methods. Multi-step, linear schemes. Predictor-corrector mode. Scheme order, relationship between order and number of stages.
2. The concept of convergence, convergence of single-step schemes. Scheme consistency. Multi-step schemes; the concept of stability and strong stability. Dahlquist's convergence theorem.
Consequences of the lack of strong stability.
3. Absolute stability, the region of absolute stability. The concept of stiffness, an example of a rigid system, the stiffness coefficient.
4. Partial differential equations and approximation by finite difference schemes. The concepts of consistency, stability, and convergence of a scheme. Lax's convergence theorem for linear equations.
5. Stability of single-step schemes. Von Neumann stability condition, strong stability. Examples for first-order equations in two independent variables.
6. Examples of schemes for equations with variable coefficients. "Upwind" schemes, conservative schemes.
7. Information on methods for equations in multiple independent variables: ADI methods and finite element methods.