Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations

Stoll M., YÜCEL H.

AIMS MATHEMATICS, cilt.3, sa.1, ss.66-95, 2018 (ESCI) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 3 Sayı: 1
  • Basım Tarihi: 2018
  • Doi Numarası: 10.3934/math.2018.1.66
  • Dergi Adı: AIMS MATHEMATICS
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
  • Sayfa Sayıları: ss.66-95
  • Anahtar Kelimeler: Allen-Cahn/Cahn-Hilliard equations, fractional diffusion, contour integral method, implicit-explicit methods, discontinuous Galerkin methods, CAHN-HILLIARD EQUATION, FINITE-ELEMENT-METHOD, PARTIAL-DIFFERENTIAL-EQUATIONS, ADVECTION-DISPERSION EQUATION, REACTION-DIFFUSION EQUATIONS, IMPLICIT-EXPLICIT METHODS, FOURIER SPECTRAL METHOD, RUNGE-KUTTA METHODS, ALLEN-CAHN, ANOMALOUS DIFFUSION
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

Fractional differential equations are becoming increasingly popular as a modelling tool to describe a wide range of non-classical phenomena with spatial heterogeneities throughout the applied sciences and engineering. However, the non-local nature of the fractional operators causes essential difficulties and challenges for numerical approximations. We here investigate the numerical solution of fractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contour integral method (CIM) for computing the fractional power of a matrix times a vector. Time discretization is performed by the first-and second-order implicit-explicit schemes with an adaptive time-step size approach, whereas spatial discretization is performed by a symmetric interior penalty Galerkin (SIPG) method. Several numerical examples are presented to illustrate the effect of the fractional power.