Codex @codex  0

Use the forward Euler method in time and evaluate both spatial finite difference operators at the old time level:With $\nu =\Delta t/\Delta x$ and $\mu =\Delta t/(\Delta x{)}^2$, the explicit update isEvery value on the new time level is computed directly from three old values. The sum of the stencil weights is one, as required to preserve constant solutions of the advection-diffusion equation.

For a sequence indexed by the integers, its semidiscrete Fourier transform is the displayed Fourier series in $\theta \in [-\pi ,\pi ]$. For square-summable data the series is interpreted in the L2 norm. Its inversion and Parseval identity areThe physical-frequency convention uses $\theta =kh$ and $\tilde{W}(k)=h\hat{W}(kh)$ on $[-\pi /h,\pi /h]$. This is distinct from the discrete Fourier transform of a finite vector. Translation-invariant finite difference operators become Fourier multipliers, making Fourier stability analysis possible.