How to implement Lax-Friedrich flux splitting with WENO scheme

Question

I'm working on modeling a shock wave using the Euler equation with an advanced Equation of state and the fifth order WENO scheme. The equation are set up on the form: \begin{equation} \frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x} = 0 \end{equation} where \begin{equation} U = \begin{bmatrix}\rho\\ \rho v\\E\end{bmatrix}, F(U) = \begin{bmatrix}\rho v\\ \rho v^2 + p\\v(E+p)\end{bmatrix} \end{equation} I've formulated the WENO scheme, but have yet to get it to work. What I find difficult to implement is the Lax-Friedrich Flux splitting. Specifically determining the $\alpha$ in the equation:

\begin{equation} f^{\pm} = f(U) \pm \alpha U \end{equation}

I do know that $\alpha = \max \left|\frac{\partial f(U)}{\partial U}\right|$, but for what index range of $U$ does the equation strech when i want to determine:

\begin{equation} \hat{f}_{i+1/2} = \hat{f}_{i+1/2}^+ + \hat{f}_{i+1/2}^- \end{equation}

Answer 1

I think you mean with Lax-Friedrichs the local Lax-Friedrichs or Rusanov scheme [1], [2] where the wave speeds (left & right-going) are given by the maximum eigenvalue $\lambda_\max := \max_i | \lambda_i(\boldsymbol{U}) | \in \sigma\Big( \boldsymbol{F}'(\boldsymbol{U}) \Big)$ where $\sigma$ denotes the spectrum of the Jacobian $\boldsymbol{F}'(\boldsymbol{U})$.

Note that the Rusanov / Local Lax Friedrich scheme is an approximate Riemann solver: For a conservation/balance law with Riemannian initial data, it provides an approximate solution. In particular, the scheme depends only on the trace (face / edge) values at the cell boundary / interface. In fact, it does not care how you got them - i.e., the scheme is agnostic whether you used simply the cell average value, a linear approximation with limiters or in your case the fifth order WENO scheme to reconstruct the face / trace values.

To finally answer your question, you need to take only the trace values $\boldsymbol{U}^-, \boldsymbol{U}^+$ into account at a certain face. Then,

$$\alpha := \max \bigg\{ \max_i \big \vert \lambda_i (\boldsymbol{U}^-) \big \vert \in \sigma\Big( \boldsymbol{F}'(\boldsymbol{U}) \Big) \Big \vert_{\boldsymbol{U}^-} , \max_i \big \vert \lambda_i (\boldsymbol{U}^+) \big \vert \in \sigma\Big( \boldsymbol{F}'(\boldsymbol{U}) \Big) \Big \vert_{\boldsymbol{U}^+} \bigg\}. $$ In other words at each face $i$ you jest need the trace values $\boldsymbol{U}_i^+, \boldsymbol{U}_i^-$. In that case, your numerical flux (I think you lack the factor if $1/2$ in your formulation and confused the signs) is given by \begin{align} \hat {\boldsymbol{f}}_{i+1/2}& = \frac{1}{2} \Big( \hat {\boldsymbol{f}}_{i+1/2}^+ + \hat {\boldsymbol{f}}_{i+1/2}^- \Big) =\frac{1}{2} \Big( F(\boldsymbol{U}_i^+) - \alpha \boldsymbol{U}_i^+ + F(\boldsymbol{U}_i^-) + \alpha \boldsymbol{U}_i^- \Big) \\ &= \frac{1}{2} \Big( F(\boldsymbol{U}_i^+) + F(\boldsymbol{U}_i^-) + \alpha (\boldsymbol{U}_i^- - \boldsymbol{U}_i^+ \Big) \end{align}