论文标题
稀疏的网格实现了eikonal方程的定点快速扫描WENO方案
Sparse grid implementation of a fixed-point fast sweeping WENO scheme for Eikonal equations
论文作者
论文摘要
固定点快速扫描方法是文献中开发的一类显式迭代方法，可有效求解双曲偏微分方程（PDES）的稳态解决方案。作为其他类型的快速扫描方案，固定点快速扫描方法使用高斯式迭代和交替的扫描策略，在每个扫描顺序中同时同时覆盖双曲PDE的特征。由此产生的迭代方案具有与稳态解决方案的快速收敛速率。此外，与其他类型的快速扫描方法相比，定点快速扫描方法的优点是它们是显式的，并且不涉及任何非线性本地系统的倒数操作。因此，它们具有稳健性和灵活性，并且与高阶准确加权基本上非振荡（WENO）方案结合在一起，以求解文献中的各种双曲线PDE。对于多维非线性问题，高阶固定点快速扫描WENO方法仍然需要大量的计算成本。在此技术说明中，我们将稀疏网格技术（用于多维问题的有效近似工具）应用于固定点快速扫描WENO方法来降低其计算成本。在这里，我们专注于具有三阶精度的强大runge-kutta（RK）类型的固定点快速扫描WENO方案（Zhang etal。2006[33]），用于求解Eikonal方程，这是一类重要的静态汉密尔顿-Jacobi（H-J）方程。进行了求解多维碘方程和更通用的静态H-J方程的数值实验，以表明固定点快速扫描的WENO方案的稀疏网格计算可在精制的网格上获得大量CPU时，同时可以保持可比较的准确性和分辨率与相应的常规单个Grids上的可比性和分辨率。
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping schemes, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The resulting iterative schemes have fast convergence rate to steady state solutions. Moreover, an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve inverse operation of any nonlinear local system. Hence they are robust and flexible, and have been combined with high order accurate weighted essentially non-oscillatory (WENO) schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high order fixed-point fast sweeping WENO methods still require quite large amount of computational costs. In this technical note, we apply sparse-grid techniques, an effective approximation tool for multidimensional problems, to fixed-point fast sweeping WENO method for reducing its computational costs. Here we focus on a robust Runge-Kutta (RK) type fixed-point fast sweeping WENO scheme with third order accuracy (Zhang et al. 2006 [33]), for solving Eikonal equations, an important class of static Hamilton-Jacobi (H-J) equations. Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse grid computations of the fixed-point fast sweeping WENO scheme achieve large savings of CPU times on refined meshes, and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
