2.5.6 Linear stability analysis of donor cell (FTBS) scheme

Applying ansatz (123) to the donor cell scheme Eq. (103) gives, using Eq. (125),

$$A = 1 - \alpha \left( 1 - e^{ jk \Delta x} \right) ... ...\cos k \Delta x\right) + j\alpha \sin k \Delta x \enspace ,$$ (129)

$$\mbox{abs} \! \left( A \right) = [ 1 - 2 \underbrace{\... ... \cos k \Delta x\right)}_{\ge 0} ]^{-\frac{1}{2}} \enspace ,$$ (130)

$${\color{HIGH1color} \mbox{abs} \! \left( A \right) \le 1... ...\mbox{for} \enspace \alpha \in \left[ 0, 1 \right] \enspace .$$ (131)

The donor cell scheme is stable if the Courant-Friedrichs-Levy condition ( CFL condition) is fulfilled,

$${\color{HIGH2color} \frac{\Delta t}{\Delta x} v_{} \in \left[ 0, 1 \right] } \enspace .$$ (132)

Note: $v_{} \ge 0$ is required (for $v_{} \le 0$ use FTFS).

Note: $\mbox{abs} \! \left( A \right) < 1$ is possible: numerical viscosity

Note: $\mbox{phase} \! \left( A \right) \neq v_{} k \Delta t$: dispersion