We introduce a new scheme for solving the non-regularized Porous Medium Equation. It is mass conserving and uses only positive unknown values. To address these typically conflicting features, we employ the eXtreme Mesh deformation approach (X-Mesh), specifically designed for problems involving sharp interfaces. The method ensures that the interface is always meshed, even in the face of complex topological changes, without the need for remeshing or altering the mesh topology. We illustrate the effectiveness of the approach through various numerical experiments.

翻译：本文提出了一种求解非正则化多孔介质方程的新方案。该方案满足质量守恒，且仅使用正的未知量数值。为兼顾这些通常相互冲突的特性，我们采用了专为处理尖锐界面问题而设计的eXtreme Mesh变形方法（X-Mesh）。该方法确保界面始终被网格覆盖，即使面对复杂的拓扑变化也无需重新划分网格或改变网格拓扑结构。我们通过多种数值实验验证了该方法的有效性。