<p>This paper presents a positivity-preserving numerical scheme for a class of nonlinear reaction-diffusion-advection algae-mussel model under Danckwerts boundary conditions. To avoid the oscillations arising in approximation of advection-diffusion models by classical numerical methods, an improved numerical method is constructed which can preserve the positivity of the original model without stepsize restriction. Furthermore, it is demonstrated that the numerical solutions converge to the exact solutions with second-order spatial accuracy. The discrete system is then linearized around the numerical semi-trivial steady state. The threshold stability of the resulting system is investigated and a stability threshold is proposed as the stability criterion and denoted as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{R}_{\varvec{0}}^{\varvec{\Delta } \varvec{x}}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold">Δ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. It is proved that the numerical semi-trivial steady state is asymptotically stable while <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{R}_{\varvec{0}}^{\varvec{\Delta } \varvec{x}}\varvec{<}\varvec{1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold">Δ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> </mrow> </msubsup> <mrow> <mo mathvariant="bold"><</mo> </mrow> <mrow> <mn mathvariant="bold">1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and unstable while <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{R}_{\varvec{0}}^{\varvec{\Delta } \varvec{x}}\varvec{>}\varvec{1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> <mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold">Δ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> </mrow> </msubsup> <mrow> <mo mathvariant="bold">></mo> </mrow> <mrow> <mn mathvariant="bold">1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, the proposed numerical stability threshold preserves the qualitative properties of the exact stability threshold. Finally, the conclusions are illustrated by numerical experiments.</p>

Threshold stability analysis of an unconditionally positivity-preserving method for a nonlinear reaction-diffusion-advection algae-mussel model under Danckwerts boundary conditions

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发表日期：

2025年5月24日

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Yanhua Lang,
Xing Liu,
Wenli Li,
Huizi Yang

摘要

This paper presents a positivity-preserving numerical scheme for a class of nonlinear reaction-diffusion-advection algae-mussel model under Danckwerts boundary conditions. To avoid the oscillations arising in approximation of advection-diffusion models by classical numerical methods, an improved numerical method is constructed which can preserve the positivity of the original model without stepsize restriction. Furthermore, it is demonstrated that the numerical solutions converge to the exact solutions with second-order spatial accuracy. The discrete system is then linearized around the numerical semi-trivial steady state. The threshold stability of the resulting system is investigated and a stability threshold is proposed as the stability criterion and denoted as $\varvec{R}_{\varvec{0}}^{\varvec{\Delta } \varvec{x}}$ $R_{0}^{Δ x}$ . It is proved that the numerical semi-trivial steady state is asymptotically stable while $\varvec{R}_{\varvec{0}}^{\varvec{\Delta } \varvec{x}}\varvec{<}\varvec{1}$ $R_{0}^{Δ x} < 1$ , and unstable while $\varvec{R}_{\varvec{0}}^{\varvec{\Delta } \varvec{x}}\varvec{>}\varvec{1}$ $R_{0}^{Δ x} > 1$ . Moreover, the proposed numerical stability threshold preserves the qualitative properties of the exact stability threshold. Finally, the conclusions are illustrated by numerical experiments.