Invariant meshless discretization schemes

A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a way of associating invariant functions to non-invariant functions. An invariant meshless approximation of a nonlinear diffusion equation is constructed. Comparative numerical tests with a non-invariant meshless scheme are presented. These tests yield that invariant meshless schemes can lead to substantially improved numerical solutions compared to numerical solutions generated by non-invariant meshless schemes.

💡 Research Summary

The paper introduces a systematic procedure for constructing meshless discretization schemes that preserve the Lie‑group symmetries of the differential equations they approximate. Traditional mesh‑based methods require the generation and maintenance of grids, which can be computationally expensive and cumbersome, especially for problems with moving boundaries or complex geometries. Meshless methods avoid explicit grids by using only a set of scattered points, but most existing meshless schemes are not symmetry‑preserving; this can lead to numerical solutions that violate conserved quantities or exhibit artificial anisotropy.

The authors base their approach on the theory of equivariant moving frames. A moving frame is a choice of group parameters that normalizes a set of variables with respect to a prescribed group action. By solving the normalization equations, one obtains explicit expressions for the group parameters in terms of the original variables. Substituting these expressions into any non‑invariant quantity yields its invariant counterpart—a process called invariantization. The paper shows how to apply this machinery to meshless approximations: first, a set of points ( {x_i} ) and associated function values ( {u_i} ) is selected; then a local Taylor‑type expansion is built using weighted least‑squares or radial‑basis‑function (RBF) interpolation, which provides the raw (non‑invariant) discrete derivatives. The symmetry group of the underlying PDE is identified, a moving frame is constructed by imposing convenient normalization conditions (e.g., fixing the value of the independent variable, the dependent variable, and certain derivatives at a reference point), and finally the raw derivative formulas are invariantized. The result is a meshless stencil that transforms equivariantly under the Lie group, guaranteeing that the discrete scheme respects the same symmetries as the continuous model.

To demonstrate the methodology, the authors focus on a nonlinear diffusion equation \