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

Download

10.3934Math.2018.1.66.pdf

Date

2018-01-01

Author

Stoll, Martin
Yücel, Hamdullah

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

240
views

288
downloads

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.

Subject Keywords

Allen-Cahn/Cahn-Hilliard equations, Fractional diffusion, Contour integral method, Implicit-explicit methods, Discontinuous Galerkin methods

URI

https://hdl.handle.net/11511/30660

Journal

AIMS MATHEMATICS

DOI

https://doi.org/10.3934/math.2018.1.66

Collections

Graduate School of Applied Mathematics, Article

Suggestions

OpenMETU
Core

Numerical studies of Korteweg-de Vries equation with random input data Üreten, Mehmet Alp; Yücel, Hamdullah; Uğur, Ömür; Department of Scientific Computing (2018) Differential equations are the primary tool to mathematically model physical phenomena in industry and natural science and to gain knowledge about its features. Deterministic differential equations does not sufficiently model physically observed phenomena since there exist naturally inevitable uncertainties in nature. Employing random variables or processes as inputs or coefficients of the differential equations yields a stochastic differential equation which can clarify unnoticed features of physical event...
Time-space fractional governing equations of one-dimensional unsteady open channel flow process: Numerical solution and exploration Ercan, Ali; Kavvas, M. Levent (2017-07-01) Although fractional integration and differentiation have found many applications in various fields of science, such as physics, finance, bioengineering, continuum mechanics, and hydrology, their engineering applications, especially in the field of fluid flow processes, are rather limited. In this study, a finite difference numerical approach is proposed to solve the time-space fractional governing equations of 1-dimensional unsteady/non-uniform open channel flow process. By numerical simulations, results of...
Formal Modelling Approaches to Complexity Science in Roman Studies: A Manifesto Brughmans, Tom; Hanson, John; Mandich, Matthew; Romanowska, Iza; Rubio-Campillo, Xavi; Carrignon, Simon; Collins-Elliott, Stephen; Crawford, Katherine; Daems, Drıes; Fulminante, Francesca; De Haas, Timon; Kelly, P; Del Carmen Moreno Escobar, Maria; Paliou, Eleutheria; Prignano, Luce; Ritondale, Manuele (2019-07-01) Complexity science refers to the theoretical research perspectives and the formal modelling tools designed to study complex systems. A complex system consists of separate entities interacting following a set of (often simple) rules that collectively give rise to unexpected patterns featuring vastly different properties than the entities that produced them. In recent years a number of case studies have shown that such approaches have great potential for furthering our understanding of the past phenomena expl...
Parallel preconditioning techniques for numerical solution of three dimensional partial differential equations Sivas, Abdullah Ali; Manguoğlu, Murat; Department of Scientific Computing (2016) Partial differential equations are commonly used in industry and science to model observed phenomena and gain insight regarding phenomena or solve related problems. Recently three dimensional partial differential equations started to become more and more essential and popular. Numerical solution of these problems usually is composed of two steps; discretization with some scheme and solving resulting sparse linear system which is large and usually ill-conditioned. Large size encourages usage of iterative sol...
Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation UZUNCA, MURAT; Karasözen, Bülent; Sariaydin-Filibelioglu, Ayse (2015-09-18) We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with nondivergence-free velocity fields. Numerical simulations for convection dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.

Citation Formats

M. Stoll and H. Yücel, “Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations,” AIMS MATHEMATICS, pp. 66–95, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30660.