Linear case

In the linear case, where f(u) = Au, and A is a constant,^[2]

$${\displaystyle u_{i}^{n+1}=u_{i}^{n}-{\frac {\Delta t}{2\Delta x}}A\left[u_{i+1}^{n}-u_{i-1}^{n}\right]+{\frac {\Delta t^{2}}{2\Delta x^{2}}}A^{2}\left[u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}\right].}$$

Here ${\displaystyle n}$ refers to the ${\displaystyle t}$ dimension and ${\displaystyle i}$ refers to the ${\displaystyle x}$ dimension. This linear scheme can be extended to the general non-linear case in different ways. One of them is letting

$${\displaystyle A(u)=f'(u)={\frac {\partial f}{\partial u}}}$$

Non-linear case

The conservative form of Lax-Wendroff for a general non-linear equation is then:

$${\displaystyle u_{i}^{n+1}=u_{i}^{n}-{\frac {\Delta t}{2\Delta x}}\left[f(u_{i+1}^{n})-f(u_{i-1}^{n})\right]+{\frac {\Delta t^{2}}{2\Delta x^{2}}}\left[A_{i+1/2}\left(f(u_{i+1}^{n})-f(u_{i}^{n})\right)-A_{i-1/2}\left(f(u_{i}^{n})-f(u_{i-1}^{n})\right)\right].}$$

where ${\displaystyle A_{i\pm 1/2}}$ is the Jacobian matrix evaluated at ${\textstyle {\frac {1}{2}}(u_{i}^{n}+u_{i\pm 1}^{n})}$.

Richtmyer method

What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for f(u(x, t)) at half time steps, t_{n + 1/2} and half grid points, x_{i + 1/2}. In the second step values at t_{n + 1} are calculated using the data for t_n and t_{n + 1/2}.

First (Lax) steps:

$${\displaystyle u_{i+1/2}^{n+1/2}={\frac {1}{2}}(u_{i+1}^{n}+u_{i}^{n})-{\frac {\Delta t}{2\,\Delta x}}(f(u_{i+1}^{n})-f(u_{i}^{n})),}$$

$${\displaystyle u_{i-1/2}^{n+1/2}={\frac {1}{2}}(u_{i}^{n}+u_{i-1}^{n})-{\frac {\Delta t}{2\,\Delta x}}(f(u_{i}^{n})-f(u_{i-1}^{n})).}$$

Second step:

$${\displaystyle u_{i}^{n+1}=u_{i}^{n}-{\frac {\Delta t}{\Delta x}}\left[f(u_{i+1/2}^{n+1/2})-f(u_{i-1/2}^{n+1/2})\right].}$$

MacCormack method

Another method of this same type was proposed by MacCormack. MacCormack's method uses first forward differencing and then backward differencing:

First step:

$${\displaystyle u_{i}^{*}=u_{i}^{n}-{\frac {\Delta t}{\Delta x}}(f(u_{i+1}^{n})-f(u_{i}^{n})).}$$

Second step:

$${\displaystyle u_{i}^{n+1}={\frac {1}{2}}(u_{i}^{n}+u_{i}^{*})-{\frac {\Delta t}{2\Delta x}}\left[f(u_{i}^{*})-f(u_{i-1}^{*})\right].}$$

Alternatively, First step:

$${\displaystyle u_{i}^{*}=u_{i}^{n}-{\frac {\Delta t}{\Delta x}}(f(u_{i}^{n})-f(u_{i-1}^{n})).}$$

Second step:

$${\displaystyle u_{i}^{n+1}={\frac {1}{2}}(u_{i}^{n}+u_{i}^{*})-{\frac {\Delta t}{2\Delta x}}\left[f(u_{i+1}^{*})-f(u_{i}^{*})\right].}$$