Unconditionally stable methods for gradient flow using Convex Splitting Runge–Kutta scheme

Jaemin Shin, Hyun Geun Lee, June Yub Lee

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


We propose a Convex Splitting Runge–Kutta (CSRK) scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner. The key feature of the scheme is a combination of a convex splitting method and a specially designed multi-stage two-additive Runge–Kutta method. Our methods are high order accurate in time and assure the gradient (energy) stability for any time step size. We provide detailed proof of the unconditional energy stability and present issues on the practical implementations. We demonstrate the accuracy and stability of the proposed methods using numerical experiments of the Cahn–Hilliard equation.

Original languageEnglish
Pages (from-to)367-381
Number of pages15
JournalJournal of Computational Physics
StatePublished - 15 Oct 2017


  • Cahn–Hilliard equation
  • Convex splitting
  • Energy stability
  • Gradient flow
  • Gradient stability
  • Phase-field model


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