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A structure-preserving upwind DG scheme for a degenerate phase-field tumor model

Daniel Acosta-Soba, Francisco Guillén-González, J. Rafael Rodríguez Galván

TL;DR

A modification of the phase-field tumor growth model given in [26] that leads to bounded, more physically meaningful, volume fraction variables is presented and an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model is developed.

Abstract

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.

A structure-preserving upwind DG scheme for a degenerate phase-field tumor model

TL;DR

A modification of the phase-field tumor growth model given in [26] that leads to bounded, more physically meaningful, volume fraction variables is presented and an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model is developed.

Abstract

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.
Paper Structure (11 sections, 13 theorems, 79 equations, 15 figures)

This paper contains 11 sections, 13 theorems, 79 equations, 15 figures.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Modified tumor model
  4. Numerical approximation
  5. Time-discrete scheme
  6. Fully discrete scheme
  7. Definition of $a_h^{\text{upw}}(\cdot;\cdot,\cdot)$
  8. Properties of the fully discrete scheme
  9. Numerical experiments
  10. Three tumors aggregation
  11. Irregular tumor growth

Key Result

Proposition 2.3

Given $u_0,v_0\in[0,1]$, any solution $(u,\mu_u,n)$ of the model problema:van_der_zee_form_var_def satisfies that $u(t)$ and $n(t)$ are bounded in $[0,1]$ for a.e. $t\in(0,T)$.

Figures (15)

  • Figure 1: Mesh used for domain discretization.
  • Figure 2: Initial conditions for test \ref{['sec:numer-experiments_1']} ($u_0$ left, $n_0$ right).
  • Figure 3: Tumor and nutrients for test \ref{['sec:numer-experiments_1']} with $\chi_0=0$ at different time steps.
  • Figure 4: Tumor and nutrients for test \ref{['sec:numer-experiments_1']} with $\chi_0=10$ at different time steps.
  • Figure 5: Pointwise bounds of the approximations for test \ref{['sec:numer-experiments_1']} with $\chi_0=0$ ($u$ left, $n$ right).
  • ...and 10 more figures

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 17 more