Abstract
Fractional differential equations are becoming increasingly popular as a modelling tool todescribe a wide range of non-classical phenomena with spatial heterogeneities throughout the appliedsciences and engineering. However, the non-local nature of the fractional operators causes essentialdifficulties and challenges for numerical approximations. We here investigate the numerical solution offractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contourintegral method (CIM) for computing the fractional power of a matrix times a vector. Time discretizationis performed by the first-and second-order implicit-explicit schemes with an adaptive time-stepsize 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 fractionalpower.
Highlights
IntroductionFractional models, in which a standard time or space differential operator are replaced by a corresponding fractional operator, have gained considerable popularity and importance during the last few decades, fractional calculus is an old topic in mathematics, see [1] for historical notes
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
Summary
Fractional models, in which a standard time or space differential operator are replaced by a corresponding fractional operator, have gained considerable popularity and importance during the last few decades, fractional calculus is an old topic in mathematics, see [1] for historical notes. In recent years, a number of successful numerical approaches for fractional differential equations have been considered such as finite difference methods [7, 8, 9, 10, 11], spectral methods [12, 13], finite element methods [14, 15, 16, 17], and discontinuous Galerkin methods [18, 19].
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