A Hybrid Finite Difference-RBF Method with Polynomial Approach and an Application to MHD Flow

This study presents a new approach to solving magnetohydrodynamic (MHD) flow problems in complex geometries using a polynomial-based Radial Basis Function-Generated Finite Difference (RBF-FD) method within a non-overlapping domain decomposition framework. It partitions the domain, specifically an L-shaped cavity with a single lid-driven, into simpler subregions where classical finite difference methods are applied, and employs the method RBF-FD at the interface points. Unlike traditional RBF approaches that require mostly shape parameter optimization, this study uses a polynomial basis function to determine derivative weights. It validates the method on benchmark lid-driven cavity problems and extends it to analyze MHD flows under various magnetic field strengths $M\\in\\{10,50,100\\}$ and orientations $\\alpha\\in\\{0^\\circ,45^\\circ,90^\\circ,135^\\circ,180^\\circ\\}$. The computational results illustrate the influence of magnetic field angle and cavity aspect ratio $\\left(h_1,h_2\\right)$ on vortex formation, revealing complex bifurcation behaviors unique to L-shaped geometries.