7.3.2 Sequential solving

In the solving process, we consider (7.22) as a sum of equations describing the convection and such equations including the viscous properties. Both systems are considered separately in the first attempt. $$\begin{eqnarray} \frac{1}{2} \partial_t \begin{pmatrix}\varrho \ \varrho u... ...tau_{22}v + k\partial_y\Theta} \end{pmatrix} &=& 0 \qquad\qquad \end{eqnarray}$$

Equations (7.23) are the Euler equations considered in section 7.2 (only differ by factor  $\tfrac{1}{2}$). There discretization was done by finite volumes in space and explicit in time. The solving in each time step can be done by algorithm 7.8 and results in a piecewise constant discrete solution $U_j^{K+1/2}$ for the $j$-th volume element in time step $k+1$.

Equations (7.24) describe the time dependent stress state of the fluid and includes second derivates of those functions. Therefore, we use FEM for the discretization in space resulting in a piecewise linear solution $u_h^{K+1}$ in time step $k+1$ when linear test functions are used.

Figure 7.5: Dual mesh

For an interaction of the discrete equations (7.23) and (7.24) we need a dual meshing of the domain by using finite volumes and finite elements ( ). One possible meshing [FFLMW97,FFLM97,FFLMW97] is given in Fig. 7.5.

When using test functions (on triangles) and definitions

$$:=$$

$$:=$$ variational formulation of elasticity operator

$$:=$$

with $g$ from (7.18) we are able to formulate the following three approaches for solving the coupled systems (7.23) and (7.24) in each time step :

1. Inviscid (7.23) - viscous (7.24) operator splitting
   
   
   
   
   Expressing the 4 components of and by FE basis functions , i.e.,
   

   $$=$$

   $$=$$

   
   
   we can write the third line in (7.25) as system of equations
   
   
   with $A$ denoting the elasticity matrix and $M$ as mass matrix, respectively. If the basis function or the integration rule are proper chosen then $M$ is a diagonal matrix and easy to invert. The definition of was already done in such a way and is equivalent to a lumping of the mass matrix.
   Usually, holds so that the requirements on the interpolation in lines 2 and 4 are fulfilled automatically.
2. Explicit scheme
   
   When applying bilinear form we can write a time step in (7.23) as
   and therefore we can express (7.22) in one discrete equation :
   Defining $B$ as matrix of bilinear form we have to solve the system of equations
   
   

3. Semi-implicit scheme
   
   In difference to the explicit scheme, we use here the actual solution $u_h^{K+1}$ in the bilinear form .
   The resulting system of equations
   
   
   is quite costly to solve because of the fact that the pattern of $A$ is not a diagonal matrix as all. Consequently, the same holds for . The advantage of the semi-implicit scheme consists in the opportunity to choose larger time steps $\tau$. If one can apply fast iterative solvers (taking into account properties of ) or has time independent matrices $A$ and $M$ (factor matrix ) then the overall solution time should be smaller in comparison to the explicit scheme.

2cm