AI 中文总结
该研究提出一种移动域带Robin边界的对流-扩散方程的笛卡尔网格方法，避免重网格化与切割单元，收敛性与计算效率良好，经数值例子验证有效性。
AI 中文摘要
我们开发了一种移动域上带有Robin边界条件的对流-扩散方程的笛卡尔网格方法。该移动域问题被重新表述为一个盒子上的界面问题，在移动界面上引入未知密度以满足Robin条件。体积方程在笛卡尔网格上采用单元中心有限差分格式离散，而界面校正则通过界面附近窄带内的局部问题获得。所得方法仅需要适度的计算几何，避免了重新网格化和切割单元，且与几何多重网格和无矩阵GMRES兼容。GMRES迭代次数基本与网格尺寸无关，计算成本随体积自由度数量线性缩放。对于一维格式，证明了时间一阶、空间二阶的收敛性。一维和二维的数值例子，包括 manufactured解和无精确解的主动输运问题，验证了该方法的准确性和效率。
英文摘要
We develop a Cartesian grid method for advection--diffusion equations with Robin boundary conditions on moving domains. The moving-domain problem is reformulated as an interface problem on a box, with an unknown density introduced on the moving interface to enforce the Robin condition. The bulk equation is discretized by a cell-centered finite-difference scheme on the Cartesian grid, while interface corrections are obtained from local problems in a narrow band around the interface. The resulting method requires only modest computational geometry, avoids remeshing and cut cells, and is compatible with geometric multigrid and matrix-free GMRES. The GMRES iteration count is essentially independent of the mesh size, and the computational cost scales linearly with the number of bulk degrees of freedom. For the one-dimensional scheme, first-order convergence in time and second-order convergence in space are proved. Numerical examples in one and two dimensions, including manufactured solutions and an active transport problem without an exact solution, demonstrate the accuracy and efficiency of the method.
