A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains
A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains
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DOI：
10.1007/s10915-023-02177-7
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发表时间：
2023-03
影响因子：
2.5
通讯作者：
Chuan Li;Y. Ren;Guangqing Long;Eric Boerman;Shan Zhao
中科院分区：
文献类型：
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作者：
Chuan Li;Y. Ren;Guangqing Long;Eric Boerman;Shan Zhao
本文提出了一种新的直角坐标网格有限差分格式，用于求解二维和三维的含不规则区域和Robin边界条件的抛物型初边值问题。在空间离散化中，利用光线投射匹配界面和边界（MIB）方法沿着法线方向强制执行不同类型的边界条件，包括Dirichlet，Neumann，Robin及其混合组合，以在不规则域之外生成必要的虚拟值。这允许在不同的边界位置处精确地近似导数的跳跃，从而可以在所有笛卡尔节点处校正四阶中心差分。通过将这种校正作为额外的未知量，可以保持拉普拉斯算子的有限差分离散化的阶数。此外，通过构造不同类型的不规则和角点的校正，所提出的增强MIB（AMIB）方法可以适应复杂的几何形状，同时保持空间的四阶精度。在时间离散中，采用标准的Crank-Nicolson格式，该格式在时间上是二阶的，并且是无条件稳定的。此外，一个快速正弦变换加速算法被用来有效地反演离散拉普拉斯算子，使得增广线性系统在每个时间步可以求解的复杂性，其中N代表总的空间自由度。所提出的AMIB方法的精度，稳定性和效率进行了数值验证，考虑各种抛物问题的二维和三维。
In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method is utilized to enforce different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mixed combinations, along the normal direction to generate necessary fictitious values outside the irregular domain. This allows accurate approximations of jumps in derivatives at various boundary locations so that the fourth-order central difference can be corrected at all Cartesian nodes. By treating such corrections as additional unknowns, the order of finite difference discretization of the Laplacian operator can be preserved. Moreover, by constructing corrections for different types of irregular and corner points, the proposed augmented MIB (AMIB) method can accommodate complicated geometries, while maintaining the fourth order of accuracy in space. In temporal discretization, the standard Crank–Nicolson scheme is employed, which is second-order in time and unconditionally stable. Furthermore, a Fast Sine Transform acceleration algorithm is employed to efficiently invert the discrete Laplacian, so that the augmented linear system in each time step can be solved with a complexity of, whereNstands for the total spatial degree-of-freedom. The accuracy, stability and efficiency of the proposed AMIB method are numerically validated by considering various parabolic problems in two and three dimensions.