A Hybrid Collocation Method for Long-Time Simulation of Heat Conduction in Anisotropic Functionally Graded Materials

This study proposes a hybrid collocation approach for simulating heat conduction problems in anisotropic functionally graded materials over extended time intervals. In this approach, the Krylov deferred correction (KDC) scheme is employed for the temporal discretization of dynamic problems, featuring a novel numerical implementation designed to ensure the precise satisfaction of boundary conditions. The localized radial basis function (LRBF) collocation method is modified and utilized to solve the resulting boundary value problems. A new radial basis function is developed and combined with an optimization strategy for the distribution of source points to enhance the performance of the LRBF scheme. This method synergizes the KDC technique, which supports large time step sizes, with the LRBF collocation method, characterized by its truly meshless nature, to address dynamic problems over long durations. Additionally, the coefficient matrix produced by the LRBF method is sparse and depends solely on the spatial distances between collocation points and source points, which is advantageous for long-term simulations. Numerical simulations spanning thousands of time steps demonstrate the accuracy, stability, and convergence of the hybrid approach. The developed numerical framework shows significant improvements over existing methods, particularly in handling dynamic problems with substantial temperature variations.