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The rotating cone problem
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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:
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Time-stepping algorithm
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Formulation for spatial discretization
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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.
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.
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