Trefftz Difference Schemes on Irregular Stencils

TL;DR

This paper introduces a generalized Trefftz-based difference scheme using least-squares fitting on irregular stencils, enhancing robustness and applicability to complex 2D and 3D physical problems.

Contribution

It extends the FLAME method by allowing least-squares fitting on irregular stencils, enabling more flexible and robust numerical schemes for differential equations.

Findings

Accuracy is maintained with irregular stencils.

Robustness of FLAME schemes is improved.

Effective in simulating electromagnetic and wave phenomena.

Abstract

The recently developed Flexible Local Approximation MEthod (FLAME) produces accurate difference schemes by replacing the usual Taylor expansion with Trefftz functions -- local solutions of the underlying differential equation. This paper advances and casts in a general form a significant modification of FLAME proposed recently by Pinheiro & Webb: a least-squares fit instead of the exact match of the approximate solution at the stencil nodes. As a consequence of that, FLAME schemes can now be generated on irregular stencils with the number of nodes substantially greater than the number of approximating functions. The accuracy of the method is preserved but its robustness is improved. For demonstration, the paper presents a number of numerical examples in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering of electromagnetic (acoustic) waves, and wave propagation in a photonic crystal. The examples explore the role of the grid and stencil size, of the number of approximating functions, and of the irregularity of the stencils.