An accelerated efficient meshless coupled algorithm for 2D/3D Cahn-Hilliard tumor-growth model with time-memory effect

In this work, a fast and accurate coupled meshless scheme is proposed to solve the time fractional Cahn–Hilliard (TFCH) equation, and for the first time, it is extended to predict 2D/3D tumor cell evolution governed by a time-memory tumor growth model. The scheme is developed by: (a) decomposing the spatial derivative into two second-order derivatives and approximating them using corrective smoothed particle hydrodynamics (CSPH); (b) applying a fast L₂ scheme to discretize the Caputo-type time-fractional derivative; (c) introducing a ghost particle technique to handle Neumann boundary conditions. Additionally, multi-CPU MPI parallelization is implemented to accelerate 3D computations. In numerical experiments, firstly, the method’s second-order convergence is demonstrated by solving a 2D TFCH problem. Secondly, its reliability, flexibility, and energy dissipation in capturing phase separation under time-memory effects are verified through a 2D kissing bubble simulation, with computational efficiency compared to the standard L₂ scheme. Thirdly, the method is used to predict long-time-memory 2D tumor cell evolution and compared with finite difference method (FDM) results. Finally, 3D multi-tumor cell evolution under time-memory effects is simulated to showcase the method’s extended capability. All results confirm that the proposed meshless method is highly efficient, accurate, and robust for modeling multi-dimensional time-memory tumor growth.