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Equations governing transient convection problem are
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Note that boundary conditions are only imposed on the inflow boundary.
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We will now only consider linear problems, so that the convective
term can be written as
f(u) = a u
with a
independent of the solution u.
When the convection velocity a
is divergence-free the problem can be rewritten as
The main feature of this equation is that space and time are linked by the characteristics.
To compute a numerical solution, a double discretization has to be performed
(in space and in time) and to obtain accurate solutions it cannot be done anyhow.
Stability of numerical methods depends on the Courant number
 ,
that links spatial and time discretization.
It is also important the order in which discretizations are performed. We will first carry out
time discretization using one of the following time-stepping algorithms:
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 - family methods (such as Euler, Crank-Nicolson...) |
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Lax-Wendroff method |
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Leap-Frog method |
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Taylor-Galerkin methods (higher order) |
Then, we use the finite element method for solving the steady problem
posed in each time step.
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Propagation of a cosine profile
Traveling wave package
The rotating cone problem
Steady Transport