In this work, we present a modification of the phase-field tumor growth model given in [26] that leads to bounded, more physically meaningful, volume fraction variables. In addition, we develop an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model. Finally, some computational tests in accordance with the theoretical results are introduced. In the first test, we compare our DG scheme with the finite element (FE) scheme related to the same time approximation. The DG scheme shows a well-behavior even for strong cross-diffusion effects in contrast with FE where numerical spurious oscillations appear. Moreover, the second test exhibits the behavior of the tumor-growth model under different choices of parameters and also of mobility and proliferation functions.

翻译：本文对文献[26]中的相场肿瘤生长模型进行了改进，使其体积分数变量具有有界性和更强的物理意义。此外，我们发展了一种保持连续模型质量守恒、逐点界和能量稳定性的迎风间断伽辽金(DG)格式。最后，引入了与理论结果一致的计算实验。在第一个实验中，我们将所提出的DG格式与相同时间近似下的有限元(FE)格式进行了比较。结果表明，即使在强交叉扩散效应下，DG格式仍具有良好的表现，而FE格式则会出现数值伪振荡。第二个实验展示了肿瘤生长模型在不同参数以及迁移率和增殖函数选择下的行为特征。