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Denis Serre
Denis Serre
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A Lax-Friedrichs (LF) type of flux for a conservation law $\partial_tU+\partial_xf(U)=0$ is given by \begin{align} F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^+ - U^-) \end{align} where for a classical LF flux in one dimension, $\lambda = \Delta t / \Delta x$. I understand that the motivation comes from trying to write the central flux scheme \begin{align} U^{n+1} = U^{n} - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} by \begin{align} U^{n+1} = \frac{1}{2}\left(U^n_{i+1} + U^n_{i-1} \right) - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} which then covers a wider domain than before and hence is more stable. The step of replacing the value at a point by average seems like a small trick. I am curious what is the idea behind such scheme, and if we can arrive at the scheme by writing a modified version of PDE and proceeding along more natural path.

A Lax-Friedrichs (LF) type of flux for a conservation law is given by \begin{align} F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^+ - U^-) \end{align} where for a classical LF flux in one dimension, $\lambda = \Delta t / \Delta x$. I understand that the motivation comes from trying to write the central flux scheme \begin{align} U^{n+1} = U^{n} - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} by \begin{align} U^{n+1} = \frac{1}{2}\left(U^n_{i+1} + U^n_{i-1} \right) - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} which then covers a wider domain than before and hence is more stable. The step of replacing the value at a point by average seems like a small trick. I am curious what is the idea behind such scheme, and if we can arrive at the scheme by writing a modified version of PDE and proceeding along more natural path.

A Lax-Friedrichs (LF) type of flux for a conservation law $\partial_tU+\partial_xf(U)=0$ is given by \begin{align} F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^+ - U^-) \end{align} where for a classical LF flux in one dimension, $\lambda = \Delta t / \Delta x$. I understand that the motivation comes from trying to write the central flux scheme \begin{align} U^{n+1} = U^{n} - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} by \begin{align} U^{n+1} = \frac{1}{2}\left(U^n_{i+1} + U^n_{i-1} \right) - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} which then covers a wider domain than before and hence is more stable. The step of replacing the value at a point by average seems like a small trick. I am curious what is the idea behind such scheme, and if we can arrive at the scheme by writing a modified version of PDE and proceeding along more natural path.

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Jayesh Badwaik
Jayesh Badwaik
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A mathematical motivation for Lax-Friedrich type of Numerical Fluxes

A Lax-Friedrichs (LF) type of flux for a conservation law is given by \begin{align} F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^+ - U^-) \end{align} where for a classical LF flux in one dimension, $\lambda = \Delta t / \Delta x$. I understand that the motivation comes from trying to write the central flux scheme \begin{align} U^{n+1} = U^{n} - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} by \begin{align} U^{n+1} = \frac{1}{2}\left(U^n_{i+1} + U^n_{i-1} \right) - \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big) \end{align} which then covers a wider domain than before and hence is more stable. The step of replacing the value at a point by average seems like a small trick. I am curious what is the idea behind such scheme, and if we can arrive at the scheme by writing a modified version of PDE and proceeding along more natural path.