We develop a first- and second-order finite-volume scheme to solve gradient flow equations with non-homogeneous properties, obtained in the framework of dynamical-density functional theory. The scheme takes advantage of an upwind approach for the space discretization to ensure positivity of the density under a CFL condition and decay of the discrete free energy. Our computational approach is used to study several one- and two-dimensional systems, with a general free-energy functional accounting for external fields and inter-particle potentials, and placed in non-homogeneous thermal baths characterized by anisotropic, space-dependent and time-dependent properties.

中文翻译：

---

非齐次扩散梯度流方程的有限体积格式

我们开发了一阶和二阶有限体积方案，以解决在动态密度泛函理论框架下获得的具有非均匀性质的梯度流方程。该方案利用逆风方法进行空间离散化，以确保在CFL条件下密度的正性和离散自由能的衰减。我们的计算方法用于研究几个一维和二维系统，其中一般的自由能函数考虑了外部场和粒子间的电势，并被放置在以各向异性，空间依赖和时间为特征的非均质热浴中依赖的属性。