化学污染物作用下大理石降解数值模拟中非线性扩散的水平集多重网格技术
A level-set multigrid technique for nonlinear diffusion in the numerical simulation of marble degradation under chemical pollutants
摘要：
考虑到化学污染物下大理石降解的建模，例如~在硫酸盐化过程中，我们考虑了控制非线性扩散方程及其数值逼近。为了准确地离散它，同时保持有限差分离散的简单性，域描述使用水平集技术。一个统一的笛卡尔网格被放置在一个盒子包含的域，但解决方案的定义和更新，只有在网格节点，位于域内，水平集被用来选择他们，并施加准确的边界条件。我们在时间上使用Crank-Nicolson方案，而对于空间变量，通过域内网格点的标准双精度差分方案和鬼点上的鬼单元技术（通过使用边界条件）来执行离散化。采用Newton-Raphson方法求解大型非线性系统，并对内部线性求解器采用裁剪多重网格技术。数值结果，这是非常令人满意的重建质量和计算效率方面，提出并讨论在文章的最后。
英文摘要：
Having in mind the modelling of marble degradation under chemical pollutants, e.g.~the sulfation process, we consider governing nonlinear diffusion equations and their numerical approximation.The space domain of a computation is the pristine marble object. In order to accurately discretize it while maintaining the simplicity of finite difference discretizations, the domain is described using a level-set technique. A uniform Cartesian grid is laid over a box containing the domain, but the solution is defined and updated only in the grid nodes that lie inside the domain, the level-set being employed to select them and to impose accurately the boundary conditions. We use a Crank-Nicolson scheme in time, while for the space variables the discretization is performed by a standard Finite-Difference scheme for grid points inside the domain and by a ghost-cell technique on the ghost points (by using boundary conditions). The solution of the large nonlinear system is obtained by a Newton-Raphson procedure and a tailored multigrid technique is developed for the inner linear solvers. The numerical results, which are very satisfactory in terms of reconstruction quality and of computational efficiency, are presented and discussed at the end of the paper.
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