Adaptive Multiquadric Shape Control in Differential Quadrature for Solving the Poisson Equation

The accuracy and stability of multiquadric (MQ) radial basis function methods are highly sensitive to the choice of the shape parameter, yet its optimal selection remains an open challenge. This paper investigates adaptive shape control within the Direct Differential Quadrature (DDQ) framework and introduces two novel strategies: Condition-Number-Constrained MQ (CNC-MQ) and Residual-Projection Descent MQ (RPD-MQ). Unlike classical heuristics such as spacing-based (HLOCAL-MQ), leave-one-out cross-validation (RIPPA-MQ), or fixed-shape choices, the proposed methods dynamically adjust shape parameters to regulate conditioning and reduce residual alignment. Results on the Poisson equation show that CNC-MQ and RPD-MQ significantly outperform baselines in error accuracy, achieving improvements of more than an order of magnitude, although at the expense of higher condition numbers. These preliminary findings establish adaptive MQ strategies as a promising direction for meshless PDE solvers and motivate future developments on stabilization and scalable implementation.