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<p>The randomized versions of implicit two-stage Runge–Kutta are considered to improve the rate of convergence (compared to a deterministic scheme) and stability of the approximate solution (contrasted to the solution generated by the explicit scheme). Dahlquist’s concept of A-stability is adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme is A-stable asymptotically and in probability but not in the mean-square sense.</p>

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Convergence and stability of randomized implicit two-stage Runge–Kutta schemes

  • 期刊论文

  • 发表日期：

    2025年1月16日

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  • Tomasz Bochacik,
  • Paweł Przybyłowicz,
  • Łukasz Stępień

摘要

The randomized versions of implicit two-stage Runge–Kutta are considered to improve the rate of convergence (compared to a deterministic scheme) and stability of the approximate solution (contrasted to the solution generated by the explicit scheme). Dahlquist’s concept of A-stability is adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme is A-stable asymptotically and in probability but not in the mean-square sense.