标题： 高阶无网格方法求解双曲方程 显示英文标题

标题： Higher-order meshless schemes for hyperbolic equations

摘要： 我们讨论了几个针对线性纯量双曲方程的无网格格式的阶数、效率、稳定性和正性。无网格格式是广义有限差分法（GFDMs），用于任意不规则网格，网格点之间没有连接。我们提出了一种新的类似于MUSCL的无网格格式，该格式使用中心模板，通过它可以达到任意高的阶，并将其与现有的无网格上风格式和无网格WENO格式进行了比较。新格式的稳定性通过模板中点上的上风重构得到了保证。由于在重构过程中重复使用了GFDM解，新的无网格MUSCL格式也非常高效。我们将新的MUSCL格式与多维最优阶检测（MOOD）过程结合，以避免在间断处出现虚假振荡。在一维空间中，我们的四阶MUSCL格式在稳定性和精度方面优于现有的WENO和上风格式。在二维空间中，我们的MUSCL格式的精度与现有的WENO格式相似，但明显更稳定。 显示英文摘要

摘要： We discuss the order, efficiency, stability and positivity of several meshless schemes for linear scalar hyperbolic equations. Meshless schemes are Generalised Finite Difference Methods (GFDMs) for arbitrary irregular grids in which there is no connectivity between the grid points. We propose a new MUSCL-like meshless scheme that uses a central stencil, with which we can achieve arbitrarily high orders, and compare it to existing meshless upwind schemes and meshless WENO schemes. The stability of the newly proposed scheme is guaranteed by an upwind reconstruction to the midpoints of the stencil. The new meshless MUSCL scheme is also efficient due to the reuse of the GFDM solution in the reconstruction. We combine the new MUSCL scheme with a Multi-dimensional Optimal Order Detection (MOOD) procedure to avoid spurious oscillations at discontinuities. In one spatial dimension, our fourth order MUSCL scheme outperforms existing WENO and upwind schemes in terms of stability and accuracy. In two spatial dimensions, our MUSCL scheme achieves similar accuracy to an existing WENO scheme but is significantly more stable.