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<p>This paper addresses the efficient numerical solution of large and stiff initial value problems (IVPs) arising from the space discretization of systems of nonlinear advection–diffusion-reaction partial differential equations (PDEs). To this end, we introduce a new family of linearly implicit two-step peer methods that leverage specialized preconditioners and exploit the reuse of previously computed stages. The proposed methods are constructed to ensure strong stability properties, specifically, L-stability or L(<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>)-stability with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> approaching 90<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(^{\circ }\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mo>∘</mo> </mmultiscripts> </math></EquationSource> </InlineEquation>, while maintaining low error constants. Compared to recently developed linearly implicit peer schemes, the new methods significantly reduce both the number of required function evaluations and linear systems to solve at each time step, also resulting in notably diminished error constants. Numerical experiments on nonlinear advection–diffusion-reaction problems testify to the efficiency of the new methods and confirm their accuracy and stability properties.</p>

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L(\(\theta \))-stable peer methods with reused stages for advection–diffusion-reaction problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月25日

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  • Giovanni Pagano,
  • Luis Rández

摘要

This paper addresses the efficient numerical solution of large and stiff initial value problems (IVPs) arising from the space discretization of systems of nonlinear advection–diffusion-reaction partial differential equations (PDEs). To this end, we introduce a new family of linearly implicit two-step peer methods that leverage specialized preconditioners and exploit the reuse of previously computed stages. The proposed methods are constructed to ensure strong stability properties, specifically, L-stability or L( \(\theta \) θ )-stability with \(\theta \) θ approaching 90 \(^{\circ }\) ∘ , while maintaining low error constants. Compared to recently developed linearly implicit peer schemes, the new methods significantly reduce both the number of required function evaluations and linear systems to solve at each time step, also resulting in notably diminished error constants. Numerical experiments on nonlinear advection–diffusion-reaction problems testify to the efficiency of the new methods and confirm their accuracy and stability properties.