| As a serious chronic disease,cancer(malignant tumor)poses a great threat to human health and life,with high morbidity and disability rate.The World Health Organization has listed it as one of the diseases threatening human life safety.With the continuous research of tumor biological trials and clinical trials,it is one of the hot spots of current tumor research to predict and explain the biological phenomenon of tumor growth and to make numerical simulation and analysis of the mechanism of tumor growth.In this paper,the numerical algorithm is mainly based on the thermodynamically consistent tumor growth model proposed by Oden et al.The model is a diffusion interface system with highly nonlinear coupling,consisting of the Cahn-Hilliard equation and the reaction diffusion equation coupled.A numerical algorithm for the tumor growth model is developed in this paper and its kinetic equations are analyzed,mainly as follows:First,the classical scalar auxiliary variable method,θ-scalar auxiliary variable method,squared scalar auxiliary variable method and exponential scalar auxiliary variable method are used to deal with the nonlinear terms in the tumor growth model.All of these methods are not restricted by the nonlinear form and the introduced scalar auxiliary variables are not dependent on the spatial variables.The only solvable and energy stable linear format can be obtained.Also,it is shown that after introducing scalar auxiliary variables,which satisfies the energy dissipation law.Second,the first-order semi-implicit backward Euler difference format is used to discretize time,with implicit treatment for the nonlinear terms and explicit treatment for the linear terms.The error order satisfies the first-order accuracy.The existence of uniqueness of the solution in the semi-discrete sense is demonstrated.Compared with full display and full hidden schemes,semi-hidden schemes are more efficient in solving practical problems with long time evolution.Final,the Fourier spectral method is used for the first time to discretize the space of the tumor growth model,which has periodic boundary conditions.The Fourier spectral method implicitly has periodic boundary conditions and is therefore applicable to this model.High-precision solutions can be obtained while processing small amounts of data.An effective and robust energy-stable format is constructed by investigating space-time fully discrete numerical format,satisfying the properties of mass conservation and decreasing energy. |
