Highly accurate simulations of problems including second derivatives on complex geometries are of primary interest in academia and industry. Consider for example the Navier-Stokes equations or wave propagation problems of acoustic or elastic waves. Current finite difference discretization methods are accurate and efficient on modern hardware, but they lack flexibility when it comes to complex geometries. In this work I extend the continuous summation-by-parts (SBP) framework to second derivatives and combine it with spectral-type SBP operators on Gauss-Lobatto quadrature points to obtain a highly efficient discretization (accurate with respect to runtime) of the Laplacian on complex domains. The resulting Laplace operator is defined on a grid without duplicated points on the interfaces, thus removing unnecessary degrees of freedom in the scheme, and is proven to satisfy a discrete equivalent to Green's first identity. Semi-discrete stability using the new Laplace operator is proven for the acoustic wave equation in 2D. Furthermore, the method can easily be coupled together with traditional finite difference operators using glue-grid interpolation operators, resulting in a method with great practical potential. Two numerical experiments are done on the acoustic wave equation in 2D. First on a problem with an analytical solution, demonstrating the accuracy and efficiency properties of the method. Finally, a more realistic problem is solved, where a complex region of the domain is discretized using the new method and coupled to the rest of the domain discretized using a traditional finite difference method.

翻译：针对复杂几何区域中包含二阶导数的数值问题开展高精度模拟，在学术界与工业界均具有核心价值。以纳维-斯托克斯方程或声波/弹性波传播问题为例，现有有限差分离散方法在现代硬件上虽能实现高精度与高效率，但在处理复杂几何构型时缺乏灵活性。本研究将连续求和分部（SBP）框架扩展至二阶导数，并结合基于高斯-洛巴托求积点的谱型SBP算子，从而获得复杂区域上拉普拉斯算子（基于运行时间的高精度）的高效离散格式。所得拉普拉斯算子定义于无界面重复点的网格上，有效消除了格式中的冗余自由度，并严格满足格林第一恒等式的离散等价形式。基于该新拉普拉斯算子的半离散稳定性在二维声波方程中得到了理论证明。此外，该方法可通过粘性网格插值算子与传统有限差分格式便捷耦合，展现出巨大的工程应用潜力。针对二维声波方程开展两项数值实验：首先通过具有解析解的问题验证离散方法的精度与效率特性；随后求解更具现实意义的问题——将复杂子域采用新方法离散化，并与传统有限差分方法离散的其余子域进行耦合计算。