Invariant meshless discretization schemes
For researchers in numerical methods for PDEs, this provides a way to improve accuracy by preserving symmetries, though the improvement is demonstrated on a specific nonlinear diffusion equation.
The paper introduces a method for constructing meshless discretization schemes that preserve Lie symmetries of differential equations, leading to substantially improved numerical solutions compared to non-invariant 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.