Estimating plane waves

It may seem difficult to estimate the plane slope p_x for a Lax-Wendroff filter of the form (10) because p_x appears non-linearly in the filter coefficients. However, using the analytical form of the filter, we can easily linearize it with respect to the plane slope and set up a simple iterative scheme:  

$$p_x^{(k+1)} = p_x^{(k)} + \Delta p_x^{(k)}\;,$$ (11)

where k stands for the iteration count, and $\Delta p_x^{(k)}$ is found from the linearized equation  

$$\left(\bold{A'}_x \bold{U}\right) \Delta p_x = - \bold{A}_x \bold{U}\;,$$ (12)

where $\bold{A'}_x$ is the derivative of $\bold{A}_x$ with respect to p_x. To avoid unstable division by zero when solving equation (12) for $\Delta p_x$, Adding a regularization equation  

$$\epsilon \nabla \Delta p_x \approx 0\;,$$ (13)

where $\epsilon$ is a small scalar regularization parameter, I solve system (12-13) in the least-square sense to obtain a smooth slope variation $\Delta p_x$ at each iteration. In practice, iteration process (11) quickly converges to a stable estimate of p_x.