Numerical exploration of a forward-backward diffusion equation (Q2911910)

The authors study numerically the nonlinear diffusion equation of the form NEWLINE\[NEWLINE\frac{\partial u}{\partial t}=\frac{\partial^2 \phi(u)}{\partial x^2}, \; (x,t)\in \mathbb{R}\times (0,+\infty),NEWLINE\]NEWLINE where \(u:\mathbb{R}\times (0,+\infty) \to \mathbb{R}\) and \(\phi: \mathbb{R}\to \mathbb{R}\) is a cubic-like function. This model can be obtained as singular limit of the third-order partial differential equation of pseudo-parabolic type NEWLINE\[NEWLINE\frac{\partial u}{\partial t}=\frac{\partial^2 }{\partial x^2}\bigg(\phi(u)+\varepsilon \frac{\partial u}{\partial t}\bigg), \; (x,t)\in \mathbb{R}\times (0,+\infty).NEWLINE\]NEWLINE Difference schemes for both models are proposed and their analytical properties are investigated. The numerical results with the emphasis to the propagation of transition interfaces are discussed.