
Equations governing transient convection problem are

Note that boundary conditions are only imposed on the inflow boundary.




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 divergencefree 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 timestepping algorithms:

 family methods (such as Euler, CrankNicolson...) 

LaxWendroff method 

LeapFrog method 

TaylorGalerkin methods (higher order) 
Then, we use the finite element method for solving the steady problem
posed in each time step.

