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.

翻译：本文讨论了针对线性标量双曲方程的几种无网格格式的精度、效率、稳定性及正性。无网格格式即广义有限差分法（GFDM），适用于任意不规则网格，其网格点之间不存在连接关系。我们提出了一种新型类MUSCL无网格格式，该格式采用中心模板，可实现任意高阶精度，并与现有的无网格迎风格式及无网格WENO格式进行了比较。新提出格式的稳定性通过向模板中点进行迎风重构得以保证。由于在重构中重复利用了GFDM解，新型无网格MUSCL格式同时具备高效性。我们将新MUSCL格式与多维最优阶检测（MOOD）程序相结合，以避免在间断处产生伪振荡。在一维空间中，我们的四阶MUSCL格式在稳定性和精度方面均优于现有WENO格式和迎风格式。在二维空间中，我们的MUSCL格式能达到与现有WENO格式相当的精度，且稳定性显著更优。