[Paper Review] Asymptotic behaviour for a chemo-repulsion system with quadratic production: The continuous problem and two fully discrete numerical schemes

Francisco Guillén‐González, María Ángeles Rodríguez‐Bellido+1 more|arXiv (Cornell University)|May 2, 2018

Mathematical Biology Tumor Growth3 citations

TL;DR

This paper analyzes a chemo-repulsion system with quadratic production, proving global weak-strong solutions converge exponentially to a constant state as time tends to infinity. It also introduces and analyzes two fully discrete numerical schemes—FE backward Euler and a nonlinear gradient-augmented scheme—demonstrating their solvability, unconditional energy stability, and comparable performance via numerical simulations.

ABSTRACT

In this paper we consider a repulsive chemotaxis model with quadratic production term. We analyze the large-time behavior of the global weak-strong solutions and we prove the exponential convergence to a constant state as time goes to infinity. Moreover, we study this same behaviour for two fully discrete numerical schemes associated to this model: the Finite Element (FE) backward Euler and another nonlinear scheme obtained by introducing as auxiliary variable the gradient of the chemical concentration. On the way, in order to analyze the asymptotic behaviour for the backward Euler scheme, we prove its solvability and unconditional energy-stability. Finally, we compare the numerical schemes throughout several numerical simulations.

Motivation & Objective

• To study the large-time behavior of global weak-strong solutions in a chemo-repulsion system with quadratic production.
• To establish exponential convergence of solutions to a constant state as time approaches infinity.
• To analyze the asymptotic behavior of two fully discrete numerical schemes: FE backward Euler and a nonlinear scheme using an auxiliary gradient variable.
• To prove solvability and unconditional energy-stability for the backward Euler scheme.
• To compare the performance of the two numerical schemes through extensive numerical simulations.

Proposed method

• Formulate a chemo-repulsion model with a quadratic production term in the cell density equation.
• Employ energy methods to analyze the long-time behavior of continuous solutions, proving exponential decay to equilibrium.
• Introduce a fully discrete finite element backward Euler scheme and prove its solvability and unconditional energy-stability.
• Propose a nonlinear fully discrete scheme by introducing an auxiliary variable for the chemical gradient, enhancing stability and accuracy.
• Use energy estimates and discrete Gronwall-type inequalities to analyze the asymptotic behavior of both numerical schemes.
• Conduct numerical simulations comparing solution profiles, convergence rates, and energy decay across both schemes.

Experimental results

Research questions

• RQ1 Do global weak-strong solutions of the chemo-repulsion system with quadratic production converge exponentially to a constant state as time goes to infinity?
• RQ2 Is the finite element backward Euler scheme for this system unconditionally energy-stable and solvable?
• RQ3 Can a nonlinear fully discrete scheme incorporating the chemical gradient as an auxiliary variable preserve energy stability and capture long-time behavior accurately?
• RQ4 How do the two numerical schemes compare in terms of solution quality, energy decay, and convergence to equilibrium?
• RQ5 What is the asymptotic behavior of the discrete solutions, and do they mirror the continuous system's exponential convergence?

Key findings

• Global weak-strong solutions of the continuous chemo-repulsion model with quadratic production converge exponentially to a constant state as time tends to infinity.
• The finite element backward Euler scheme is unconditionally energy-stable and unconditionally solvable, ensuring robust numerical integration.
• The nonlinear scheme with an auxiliary gradient variable preserves energy stability and accurately captures the long-time dynamics.
• Numerical simulations confirm that both schemes exhibit exponential decay of the discrete energy, aligning with the continuous system's behavior.
• The two numerical schemes show comparable performance in simulating the asymptotic behavior, with no significant degradation in accuracy or stability.