where $f_0(x)$ is the initial $f$ profile. That is, we just shift the domain. This is consistent with this PDE being a wave equation, where the solution convects across the domain without changing shape.
* O *
O O O
* O *
O * O
* O *
O O O
where
$$\overline{F} = \frac{1}{\Delta t}\int_t^{t+\Delta t}Fdt.$$If we take an explicit Euler step to $\tilde{f}_i^{n+1}$, then we can evaluate $F_i^{n+1}$ there. Then we can evaluate the above equation explicitly.
We do this, but
Summary:
For the 1-D linear wave equation, this method is the same as the Lax Wendroff method.
$\mathcal{O}(\Delta t^2)$
$\mathcal{O}(\Delta x^2)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 1$
Consistent
Stencil:
O O *
* O O
* O *
O O *
This equation is exact for the linear 1D wave equation for $c=1$.
$\mathcal{O}(\Delta t^2)$
$\mathcal{O}(\Delta x^2)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 2$
Consistent
Stencil:
* * O *
O O O *
O O O
* O *
Method | Recommend | $\mathcal{O}(\Delta t)$ | $\mathcal{O}(\Delta x)$ | Approach | Stability | Comment |
---|---|---|---|---|---|---|
FTCS | X | 1 | 2 | direct, forward in time, central in space | NO | doesn't work |
Lax | X | 1 | 2 | average $f_i^n$ | $c\le 1$ | inconsistent |
Lax Wendroff | okay | 2 | 2 | fix low order error $f_{tt}$ term | $c\le 1$ | hard for nonlinear, systems, multi-D |
MacCormack | good | 2 | 2 | time adv. with avg. RHS slope: | $c\le 1$ | good for nonlinear, systems, multi-D |
Upwind | poor | 1 | 1 | very diffusive | $c\le 1$ | diffusive, low order |
2nd Order Upwind | okay | 2 | 2 | 3 point stencil | $c\le 2$ | oscillatory |
BTCS | okay | 1 | 2 | implicit | $c\le 2$ | unphysical info. speed |