Stability and Error Analysis of Structure-Preserving Schemes for a Diffuse-Interface Tumor Growth Model

This work is concerned with a diffuse-interface (phase-field) model for tumor growth that takes into account nutrient consumption and chemotaxis. For this tumor growth model described by the nonlinear system consisting of a Cahn-Hilliard-type equation coupled with a reaction-diffusion equation, we first prove the existence of its weak solutions. Efficient first- and second-order schemes are then constructed based on the idea of the scalar auxiliary variable, which we show are not only decoupled and easy to implement, but also have the properties of mass conservation and unconditional energy stability. Furthermore, we derive rigorous error estimates for the tumor and nutrient variables of the first-order scheme. Several numerical examples are presented to validate the accuracy and stability of the proposed schemes. It is worth noting that when the scheme is equipped with an adaptive time-stepping strategy, it efficiently simulates the typical phenomena of aggregation of multiple tumors with different shapes and tumor chemotactic growth.