Lax–Wendroff method

The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite differences. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines the governing equation is evaluated at the current time.

Definition
Suppose one has an equation of the following form: $$ \frac{\partial u(x,t)}{\partial t} + \frac{\partial f(u(x,t))}{\partial x} = 0$$ 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, $$ u_i^{n+1} = u_i^n - \frac{\Delta t}{2\Delta x} A\left[ u_{i+1}^{n} - u_{i-1}^{n} \right] + \frac{\Delta t^2}{2\Delta x^2} A^2\left[ u_{i+1}^{n} -2 u_{i}^{n} + u_{i-1}^{n} \right].$$ Here $$n$$ refers to the $$t$$ dimension and $$i$$ refers to the $$x$$ dimension. This linear scheme can be extended to the general non-linear case in different ways. One of them is letting $$ A(u) = f'(u) = \frac{\partial f}{\partial u}$$

Non-linear case
The conservative form of Lax-Wendroff for a general non-linear equation is then: $$ u_i^{n+1} = u_i^n - \frac{\Delta t}{2\Delta x} \left[ f(u_{i+1}^{n}) - f(u_{i-1}^{n}) \right] + \frac{\Delta t^2}{2\Delta x^2} \left[ A_{i+1/2} \left(f(u_{i+1}^{n}) - f(u_{i}^{n})\right) - A_{i-1/2}\left( f(u_{i}^{n})-f(u_{i-1}^{n})\right) \right].$$ where $$A_{i\pm 1/2}$$ is the Jacobian matrix evaluated at $\frac{1}{2} (u^n_i + u^n_{i\pm 1})$.

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

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

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

Alternatively, First step: $$ u_{i}^{*} = u_{i}^n - \frac{\Delta t}{\Delta x}( f(u_{i}^n) - f(u_{i-1}^n) ).$$ Second step: $$ u_i^{n+1} = \frac{1}{2} (u_{i}^n + u_{i}^*) - \frac{\Delta t}{2 \Delta x} \left[ f(u_{i+1}^{*}) - f(u_{i}^{*}) \right].$$