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With this second example we'd like to show the numerical damping and the phase lag of numerical solutions.
Consider a one-dimensional pure transient convection problem with initial condition
the product of a square wave and a sinusoidal wave.
Several time-stepping algorithms are available:
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Lax-Wendroff method (second order, explicit) |
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Leap-frog method (second order, explicit) |
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Third-order explicit Taylor-Galerkin method |
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Crank-Nicolson method (second order, implicit) |
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Fourth-order implicit Taylor-Galerkin method |
Spatial discretization is performed using finite elements.
Galerkin formulation is used in all cases except for the Carey-Jiang method (in which we use
Crank-Nicolson algorithm for time and a least-squares formulation for space).
Besides, we can compute solutions using a lumped mass representation for the Lax-Wendroff
and the Crank-Nicolson algorithms.
Some results
Consider a constant convection velocity a = 1.
The Figures below show solution at time 1.5 computed using a mesh of 240 linear elements.
Time-step is chosen so that Courant number is 90% of the stability limit. For unconditionally stable
schemes, as especially good properties are obtained for C=1,we use C=0.9.
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Lax-Wendroff |
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Lax-Wendroff with lumped mass matrix |
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AVI file |
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AVI file |
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Third order explicit Taylor-Galerkin |
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Fourth order implicit Taylor-Galerkin |
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AVI file |
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AVI file |
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These figures do not show a really good performance of the methods.
Numerical solutions (except the one obtained with TG4) show important damping or phase errors.
However, the example shown above is a limit case because only 8 elements per wave are used.
The Figures below show that better results are obtained in other cases.
For instance, Carey-Jiang solution shows a remarkable reduction of numerical damping
and phase error is practically canceled for the Crank-Nicolson method.
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