Suppose one has an equation of the following form:
where x and t are independent variables, and the initial state, u(x, 0) is given.
Linear case
In the linear case, where f(u) = Au, and A is a constant,[2]
Here refers to the dimension and refers to the dimension.
This linear scheme can be extended to the general non-linear case in different ways. One of them is letting
Non-linear case
The conservative form of Lax-Wendroff for a general non-linear equation is then:
where is the Jacobian matrix evaluated at .
Jacobian free methods
To avoid the Jacobian evaluation, use a two-step procedure.
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, tn + 1/2 and half grid points, xi + 1/2. In the second step values at tn + 1 are calculated using the data for tn and tn + 1/2.
Michael J. Thompson, An Introduction to Astrophysical Fluid Dynamics, Imperial College Press, London, 2006.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 20.1. Flux Conservative Initial Value Problems". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. p. 1040.
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4896aba31477fc792026cfbdb84375689c1a6b55)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e351ecf692b371d283629cae7354a5598f3ae2f)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fe928153382db4b465a4cc76bb7b3ccc2fe2d1)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56975847eca9241721582a2ae4fae89abcd32dc3)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28b1232af80fd0fc25fae13893fd5c13fac9fcf0)