A highly efficient semi-implicit corrective SPH scheme for 2D/3D tumor growth model

In this work, a semi-implicit corrective SPH method is proposed to solve the multi-component Cahn–Hilliard equation with fourth-order derivatives, and it is further used to predict the high-dimensional tumor growth model. The scheme is motivated by: (a) the fourth-order spatial derivative is discretized continuously by a corrective SPH formula for approximating second-order derivative twice, and the Neumann boundary is imposed by a ghost technique; (b) the temporal direction is approximated by the implicit scheme, and an iterative concept is employed to handle the above implicit form; (c) the multi-CPUs MPI parallelization is adopted to reduce the computing cost. Firstly, the second-order convergence rate of the proposed method for 2D/3D equation is shown and discussed by two analytical examples, and the mass conservation and energy properties are also demonstrated. Secondly, the efficiency of the proposed approach for multi-phase separation phenomenon is illustrated in an irregular domain. Finally, the 2D/3D tumor growth evolution at a short time is predicted by the proposed method and qualitatively compared with other numerical results. The numerical experiments show that the proposed scheme for phase-separation phenomenon or tumor growth model is highly efficient and reliable.