Advanced Mathematics

Inverse Problems (IPs)

• Exposition.

Porous Medium Equations (PMEs)

PMEs: $\partial_tu - \Delta u^m + u^{-\beta}\chi_{{ u>0 }} = 0$ in $(0,\infty)\times\mathbb{R}^N$.

• Nguyen Quan Ba Hong’s. Multidimensional Degenerate Diffusion Equation with a Very Strong Absorption & A Source Term. [pdf]. [slide]. HCMUS. Jul 2018.

• Nguyen Anh Dao, Jesus Ildefonso Diáz, Quan Ba Hong Nguyen. Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term. Advanced Nonlinear Studies. [link]. [pdf]. doi: 10.1515/ans-2020-2076.

Finite Volume Methods (FVMs)

• Nguyen Quan Ba Hong’s Master Thesis. Staggered & Well-Balanced Discretization of Shallow Water Equations. [pdf]. [slide]. UR1. Jun 2019.

Turbulence Models

Navier–Stokes equations, $k$-$\varepsilon$ turbulence model(s) (several variants), $k$-$\omega$ turbulence model.

\(\nabla\cdot(\rho{\bf u}) = 0\), \(\partial_t(\rho{\bf u}) + \nabla\cdot(\rho{\bf u}\otimes{\bf u}) = \nabla\cdot[(\mu + \mu_{\rm t})\nabla{\bf u}] + {\bf Q}^{\bf u}\), \(\partial_t(\rho k) + \nabla\cdot(\rho{\bf u}k) = \nabla\cdot\left(\left(\mu + \frac{\mu_{\rm t}}{\sigma_k}\right)\nabla k\right) + P_k - \rho\varepsilon\), \(\partial_t(\rho\varepsilon) + \nabla\cdot(\rho{\bf u}\varepsilon) = \nabla\cdot\left(\left(\mu + \frac{\mu_{\rm t}}{\sigma_\varepsilon}\right)\nabla\varepsilon\right) + C_{\varepsilon1}\frac{\varepsilon}{k}P_k - C_{\varepsilon2}\rho\frac{\varepsilon^2}{k}\).