Structure-preserving upwind DG scheme for a Cahn-Hilliard-Darcy model of tumor growth

TL;DR

This paper introduces a structure-preserving numerical scheme for a coupled Cahn-Hilliard-Darcy model of tumor growth, ensuring physical consistency and robustness in simulations.

Contribution

It develops a fully discrete upwind DG scheme that preserves mass, bounds, and energy laws for a novel tumor growth model involving fluid interactions.

Findings

The scheme conserves mass and maintains bounds in simulations.

Numerical experiments confirm robustness and accuracy.

Fluid effects significantly influence tumor evolution.

Abstract

In this work, we develop a structure-preserving numerical scheme for a Cahn-Hilliard-Darcy model that describes tumor growth in a fluid-saturated porous medium. First, we derive a physically consistent model from the general framework proposed in [29] that guarantees mass conservation and pointwise bounds on the phase-field and nutrient variables, with a decreasing energy law. The resulting model couples the evolution of tumor cells via a Cahn-Hilliard equation with a diffusion equation for the nutrients thro chemotactic interactions and extends the model in [1] by introducing the effect of a surrounding fluid described by Darcy's law. Subsequently, we propose a fully discrete scheme that combines an upwind discontinuous Galerkin method in space and a convex splitting strategy in time, which inherits the fundamental properties of the continuous model: mass conservation, pointwise bounds and discrete energy law. Our theoretical analysis is accompanied by numerical experiments that demonstrate the robustness of the proposed scheme and show the influence of the surrounding fluid on the tumor evolution.