Higher-order meshless schemes for hyperbolic equations

TL;DR

This work develops high-order, positivity-preserving meshless schemes for linear hyperbolic equations on irregular grids. It introduces a MUSCL-like scheme with a central stencil and upwind mid-point reconstruction, augmented by MOOD to enforce the discrete maximum principle, and compares it to a meshless WENO approach. In 1D, the 4th-order MUSCL-MOOD scheme achieves superior stability and accuracy, while in 2D it matches WENO accuracy with notably better stability; the methods are efficiently realized by reusing the GFDM stencil in reconstructions. The study also discusses MLS-based gradient reconstruction, non-conservation caveats, Dirichlet boundary behavior, and extensive convergence, stability, conservation, and efficiency results, indicating practical potential for moving-mesh and kinetic equation applications.

Abstract

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.

Figures (11)

• Figure 1: Possible upwind stencils in two spatial dimensions. The central point of the stencil is the point at the origin. The points in blue are the neighbours $C_i$. The vector $a$ is the velocity in the linear advection equation. The shaded regions indicate possible upwind stencils. The angle $\theta_{ij}$ is the angle between the positive x-axis and the line connecting the centerpoint and the neighbour $j$.
• Figure 2: Illustration of how random grids are generated with $r \in \left[0, \frac{\Delta x}{2} \right]$.
• Figure 3: Convergence of 1D meshless schemes for smooth initial condition (left) and shock initial condition (right). The $x$-axis represents the amount of grid points. The $y$-axis plots the relative error. The numbers in the legend are the theoretical orders of the methods.
• Figure 4: Solution for the sine and shock initial condition for several schemes with $N_x = 100$. Stars indicate the location of a MOOD event at the final time step.
• Figure 5: Convergence of 2D meshless schemes for smooth initial condition (left) and shock initial condition (right). The $x$-axis represents the amount of grid points. The $y$-axis plots the relative error. The numbers in the legend are the theoretical orders of the methods.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3