# Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \Big( \dfrac{d\varepsilon_2(z)}{dz} \Big)\dfrac{\partial f}{\partial k} = 0 \end{equation} I have encountered some texts that propose using entropy functions. For example, it is mentioned that the choice of \begin{equation} h(f,z,k) = fe^{\varepsilon_1+\varepsilon_2} \end{equation} will do the job for the above equation.

How is the given function (or any other entropy function) increasing the numerical stability? What is the main idea behind using such functions?

• What do the papers that "propose using entropy functions" suggest? – Wolfgang Bangerth May 4 '16 at 19:49
• Would you add a reference to the texts that you have read ? – BrunoLevy May 5 '16 at 7:12
• math.la.asu.edu/~chris/ttsp02.pdf This paper, for example. – Maziar Noei May 5 '16 at 10:06
• Well, that's a reference. But what do they propose to use the entropy function for? – Wolfgang Bangerth May 5 '16 at 14:55
• For stabilization. When I try to directly discretize the two-dimensional advection equation, I get crazy oscillations and negative values for $f(z,k)$. I don't know how such entropy functions help with stabilizing the numerical system. – Maziar Noei May 6 '16 at 8:25