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<p>In this paper, we deal with the stability of equilibria arising from both semi-discrete and fully discrete approximations of a Gurtin-MacCamy model with infinite age span. Numerical solutions obtained through both the discontinuous Galerkin semidiscretization with piecewise constant approximation in age and the partitioned implicit-explicit Euler method are shown to replicate the stability of the equilibria. For the discontinuous Galerkin semi-discrete processes, a numerical reproduction number is introduced to derive conditions that ensure the existence of a numerical nontrivial equilibrium. Meanwhile, by linearizing the nonlinear model, the local stability of numerical equilibrium distributions is discussed. As an application to the Gurtin-MacCamy model with logistic growth, we derive a numerical threshold parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R_0^h\)</EquationSource> </InlineEquation>, which ensures the global stability of the numerical trivial equilibrium whenever <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R_0^h&lt;1\)</EquationSource> </InlineEquation> and the existence and the local stability of a unique numerical nontrivial equilibrium for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(R_0^h&gt;1\)</EquationSource> </InlineEquation>. Moreover, under the presented partitioned implicit-explicit framework, the number <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(R_0^h\)</EquationSource> </InlineEquation> is also a threshold for the fully discrete scheme for any temporal stepsize. Finally, numerical experiments are presented to validate and demonstrate the efficiency of our theoretical results.</p>

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Numerical thresholds of discontinuous Galerkin methods for the Gurtin-MacCamy model with infinite age span

  • 期刊论文

  • 发表日期：

    2026年3月13日

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  • Xiaochen Yang,
  • Zhanwen Yang

摘要

In this paper, we deal with the stability of equilibria arising from both semi-discrete and fully discrete approximations of a Gurtin-MacCamy model with infinite age span. Numerical solutions obtained through both the discontinuous Galerkin semidiscretization with piecewise constant approximation in age and the partitioned implicit-explicit Euler method are shown to replicate the stability of the equilibria. For the discontinuous Galerkin semi-discrete processes, a numerical reproduction number is introduced to derive conditions that ensure the existence of a numerical nontrivial equilibrium. Meanwhile, by linearizing the nonlinear model, the local stability of numerical equilibrium distributions is discussed. As an application to the Gurtin-MacCamy model with logistic growth, we derive a numerical threshold parameter \(R_0^h\) , which ensures the global stability of the numerical trivial equilibrium whenever \(R_0^h<1\) and the existence and the local stability of a unique numerical nontrivial equilibrium for \(R_0^h>1\) . Moreover, under the presented partitioned implicit-explicit framework, the number \(R_0^h\) is also a threshold for the fully discrete scheme for any temporal stepsize. Finally, numerical experiments are presented to validate and demonstrate the efficiency of our theoretical results.