Finite Element Methods for Flow Problems
Unsteady Transport 
  Introduction
  Steady Transport
  Unsteady Transport
Ex 1: Cosine profile
Ex 2: Wave package
Ex 3: Rotating cone
  Compressible Flow
  Unsteady Convection-Diffusion
  Incompressible Flow
The rotating cone problem

This problem is a classical test for 2D convection schemes. Consider a product cosine hill centered in (1/6,1/6) in a pure rotating convection field a = (-y, x).
The problem is solved using a uniform mesh of linear elements. There are seven different methods available:
Time-stepping algorithm
Formulation for spatial discretization
LW Lax-Wendroff Galerkin
LW-FD Lax-Wendroff Galerkin with diagonal mass matrix
TG3 third order Taylor-Galerkin Galerkin
CN Crank-Nicolson Galerkin
CN-FD Crank-Nicolson Galerkin with diagonal mass matrix
CJ Crank-Nicolson Least-Squares
TG3-2S Two-step third order (Selmin) Galerkin

The following figures show the results obtained after a complete revolution. Numerical solution has been computed using a mesh of 20x20 linear elements and 120 time steps.
As expected, Lax-Wendroff method with lumped mass matrix shows a rather poor behavior: we can appreciate some oscillations, numerical damping, and phase lag.
AVI file AVI file
The rest of the methods provide much better results. For instance, figures below show solution obtained with LW (left) TG3-2S (right) methods after two complete revolutions.
AVI file AVI file

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Compressible Flow