 # Finite Element Method (FEM)

The semi-discrete method for the calculation of the derivative with respect to time makes use of finite differences, i.e. discrete time intervals are used. Eq. 3-53 summarizes the most common approximations for time discretization, all of them being two-level methods, which means that they are based on values from two time-levels n+1 and n.

The methods for different values for are called:

The forward-Euler method is also called explicit method because the calculation of every flow and every source term of the equation to be solved is based on the old time level tn. The only variable at the time tn+1 is the variable in the vertex i. The variable is thus calculated without reference to any values of adjacent vertices (spatial derivatives), i.e. explicitly. Additional conditions have to be made for the time-step (time increment ) in order to achieve stable solutions with this method. These conditions are defined in dependency of the Peclet number. The time-step method used in RMA2 is an explicit method like as described above.

The backward-Euler method is an implicit method because the spatial derivatives, for example, are based only on the values of (adjacent) vertices of the current time level tn+1. The implicit method has no restrictions concerning the width of time-steps and its effect on the stability of the calculation as opposed to the explicit method. On the other hand, it is an iterative method, which means a higher numerical effort for the individual time-steps.

More detailed explanations concerning the solving of unsteady problems can be found in [*1] among others.  