Solving the PDE

Solving PDE (7) is done on a discretization $\widehat{\Omega}$ of the continuous domain $\Omega$.

Christensen's original approach uses successive over-relaxation (SOR) (8, pp.866-869) (2,7,6) (Algorithm 2).

As an improvement, an adaptive update scheme (SORA) is used in my implementation. In each SOR iteration $\vec{v}_{i,j,k}$ depends on the 19 values with indices

$$\kappa \in \Im :=\left\{ \left( \begin{array}{c} i j k \end... ...\right) ,\left( \begin{array}{c} i± 1 j k± 1 \end{array}\right) \right\},$$ (9)

only. An adaptive update is now introduced, using an threshold

$$\hat{r}:=\left\{ \begin{array}{ll} 0 & m=1 \overline{r}^{(m)}\... ...)}}{\overline{r}^{(m-1)}}\cdot \frac{1}{m^{2}} & otherwise \end{array}\right. ,$$ (10)

with

$$\overline{r}:=\frac{1}{X\cdot Y\cdot Z}\sqrt{\sum \left\Vert \vec{r}_{i,j,k}\right\Vert^{2}},$$ (11)

to decide, which elements to update during the iterative solution of (7) (Algorithm 3).

Another approach to solve (7) is the minimal residuum algorithm (MINRES) (2), a variant of conjugated gradients also suitable for indefinite matrices as they arise when discretizing (7) (Algorithm 4).

Finally Bro-Nielsen approach is based on folding the input force $\mathbf{f}$ (6) with the impulse response of the Navier-Stokes-operator (Section 4.3).