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]中给出的相场肿瘤生长模型进行了改进，使得体积分数变量有界且更具物理意义。此外，我们开发了一种迎风间断伽辽金（DG）格式，该格式保留了连续模型的质量守恒、逐点有界性和能量稳定性。最后，引入了一些与理论结果一致的数值测试。在第一个测试中，我们将DG格式与基于相同时间近似的有限元（FE）格式进行了比较。DG格式即使在强交叉扩散效应下也表现出良好的数值行为，而FE格式则出现了数值伪振荡。此外，第二个测试展示了肿瘤生长模型在不同参数选择下以及在不同迁移和增殖函数下的行为。