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Amplification polynomial of a multilevel finite difference scheme

For a scheme

\sum_{r, j} a_{r j} u_{m + j}^{n + r} = 0,

the Fourier ansatz gives the amplification polynomial

\sum_{r, j} a_{r j} G^{r} e^{ij θ} = 0.

Its roots are the amplification factors of that Fourier mode.

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Solution