Numerical dispersion, stability; four-point implicit (500 m)

Numerical diffusion

The leading truncation error of an odd-order scheme (e.g. first-order upwind) behaves like a physical diffusion term. For $Q_t+cQ_x=0$ with backward-space, forward-time:

$$Q_t+cQ_x=\frac{c\Delta x}{2}(1-C)\,Q_{xx}+\dots$$

The solution is smeared: peaks are lowered and sharp fronts spread out, although the true equation has no diffusion.

Numerical dispersion

For second-order (central) schemes the leading error term has an odd derivative, $\propto Q_{xxx}$. Different wavelengths then travel at different speeds, so a sharp front produces spurious oscillations (ripples ahead of or behind the front). It is a phase error, with little change in amplitude.

Stability

A scheme is stable if errors (round-off or truncation) do not grow from step to step. For explicit schemes, the von Neumann or Courant-Friedrichs-Lewy condition requires

$$C=\frac{c\,\Delta t}{\Delta x}\le1$$

Implicit schemes (and the four-point scheme with $\theta\ge0.5$) are unconditionally stable, but a large $\Delta t$ still reduces accuracy.

Four-point implicit method

$$\frac{\partial Q}{\partial t}\approx\frac{(Q_i^{n+1}-Q_i^n)+(Q_{i+1}^{n+1}-Q_{i+1}^n)}{2\Delta t}$$ $$\frac{\partial Q}{\partial x}\approx\theta\,\frac{Q_{i+1}^{n+1}-Q_i^{n+1}}{\Delta x}+(1-\theta)\,\frac{Q_{i+1}^n-Q_i^n}{\Delta x}$$

Data: $Q_i^{n+1}=75$, $Q_{i+1}^{n+1}=72$, $Q_i^n=73$, $Q_{i+1}^n=69$; $\Delta t=3600$ s, $\Delta x=500$ m, $\theta=0.55$.

Time derivative

$$\frac{\partial Q}{\partial t}=\frac{(75-73)+(72-69)}{2\times3600}=\frac{5}{7200}=6.9444\times10^{-4}\ \text{m}^3/\text{s}^2$$

Space derivative

$$\frac{\partial Q}{\partial x}=0.55\,\frac{72-75}{500}+0.45\,\frac{69-73}{500}=-0.00330+(-0.00360)=-0.00690\ \text{m}^3/\text{s per m}$$

Answer: $\partial Q/\partial t=6.9444\times10^{-4}$ m$^3$/s$^2$ and $\partial Q/\partial x=-0.00690$ m$^2$/s.