PDE 13: Buckley-Leverett equation

The equation is an example of a scalar conservation law describing two-phase flow in a porous medium (note that there is a large theory on numerical methods for conservation laws). Consider the initial value problem \[ u_t + f(u)_x = 0, \] where \[ f(u) = \frac{u^2}{u^2 + (1-u)^2}, \] with initial value \[ u(x,0) = u_0(x). \]

Google Scalar conservation law, Buckley-Leverett equation

Challenges nonlinearity, applications