• Further topics:
  
  Selection of further topics depends on the research interests of students and time available. Typically there is time left to cover only one or two of the below. Students are encouraged to convey their interests.


• Other types of methods.
  • Spectral methods
  • Collocation methods
• Physically relevant properties of methods.

  Parasitic modes; Dissipation; Dispersion; Group velocity; Conservation.

• Multistep schemes.

  Construction of higher order schemes by formal difference calculus; von Neumann analysis of multistep methods; Stability considerations via Rouché theorem; The wave and beam equations; Stability of schemes for problems second order in time; Caveats and advantages in reduction to lower order systems.

• Conservation laws.

  Nonlinear hyperbolic systems and their distributional solutions; shocks and rarefactions; Rankine-Hugoniot condition; entropy solutions; Riemann problems; Lax-Wendroff theorem; Godunov method.

• Some special equations.
  • The eikonal equation: WKB approximation; the geometrical optics limit; the fast marching method; the level set method.
  • The Stokes system: The incompressibility constraint; Instability of some obvious schemes; Projection based methods.