STABILIZED SECOND-ORDER θ-METHODS: PRESERVING ACCURACY AND ENFORCING IMAGINARY-AXIS DAMPING VIA θ = 12 + κH

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AĞGÜL M.

Evolution Equations and Control Theory, cilt.23, ss.295-307, 2026 (SCI-Expanded, Scopus)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 23
  • Basım Tarihi: 2026
  • Doi Numarası: 10.3934/eect.2026064
  • Dergi Adı: Evolution Equations and Control Theory
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.295-307
  • Anahtar Kelimeler: Crank–Nicolson, flow past a cylinder, imaginary-axis damping, second-order accuracy, undamped oscillations, θ-method
  • Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
  • Hacettepe Üniversitesi Adresli: Evet

Özet

We introduce a modified parametrization of the classical θ-method and its one-leg variant for the time integration of first-order ordinary differential equations. The standard choice θ = 1 /2 , corresponding to the Crank–Nicolson and midpoint schemes, is second-order accurate and A-stable but is known to generate persistent temporal oscillations in stiff problems and in systems with purely imaginary eigenvalues. To address this limitation, we consider a step-size-dependent parameterization θ = 1 /2 + κh, where κ > 0 is a tunable parameter and h denotes the time step. We show that any choice θ > 1 /2 yields strict damping along the imaginary axis, while second-order accuracy is retained under the scaling condition θ − 1 /2 = O(h), in particular for θ = 1 /2 + κh with bounded κ. Prescribing a nontrivial per-step damping ratio at a fixed dimensionless target frequency generally keeps θ away from 1 /2 and therefore yields a first-order method, whereas choosing the target damping as q = q(h) = 1 − O(h) preserves the second-order regime and still produces controlled cumulative decay over any fixed time interval. Explicit and asymptotic expressions are derived for selecting κ to attain a prescribed damping ratio at a target frequency, leading to a practical stabilization criterion. Numerical experiments demonstrate that the proposed θ-stab scheme suppresses spurious oscillations more effectively than the Crank–Nicolson and midpoint methods, both for a simple stiff test problem and for flow-past-a-cylinder simulations.