Full Discrete Form of Heat Conduction Equation by Meshless Fragile Points Method (FPM) and Various ODE Solvers

Potpuno diskretizovane formulacije jednačine toplotne provodljivosti na bazi bezmrežne Fragile Points metode (FPM) i različitih algoritama za rješavanje ODJ

Abstract

According to recent studies, the Fragile Points Method (FPM) has emerged as a highly promising technique for addressing problems related to heat conduction analysis. This paper further develops and extends the investigation of FPM by implementing and comparing several explicit and implicit time integration schemes for solving the resulting systems of ordinary differential equations (ODEs). In total, six fully discrete formulations of the general heat conduction equation are derived and presented. The study includes a detailed assessment of each algorithm’s computational complexity, execution time, and relative numerical accuracy, providing a comprehensive evaluation of their performance. The convergence and stability of the FPM approach were verified through multiple benchmark problems with known analytical solutions in both one and two dimensions. Furthermore, it was demonstrated that the presence of non-homogeneity and anisotropy in the analysed media does not introduce additional difficulties for the method. When compared with explicit techniques, the implicit formulations proved to be more efficient for stiff problems, achieving stable solutions while maintaining comparable computational costs.