• PDF / 3,540,719 Bytes
• 22 Pages / 439.642 x 666.49 pts Page_size
• 42 Downloads / 273 Views

DOWNLOAD

REPORT

---

Isogeometric analysis for time-fractional partial differential equations Xindi Hu1 · Shengfeng Zhu2 Received: 2 March 2019 / Accepted: 1 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019

Abstract We consider isogeometric analysis to solve the time-fractional partial differential equations: fractional diffusion and diffusion-wave equations. Traditional spatial discretization for time-fractional models include finite differences, finte elements, spectral methods, etc. A novel method-isogeometric analysis is used for spatial discretization in this paper. The traditional L1 scheme and L2 scheme are used for time discretization of our models. Isogeometric analysis has potential advantages in exact geometry representations, efficient mesh generation, h- and k- refinements, and smooth basis functions. We show stability and a priori error estimates for spatial discretization and the space-time fully discrete scheme. A variety of numerical examples in 2d and 3d are provided to verify theory and show accuracy, efficiency, and convergence of isogeometric analysis based on B-splines and non-uniform rational B-splines. Keywords Time-fractional · Isogeometric analysis · Subdiffusion · Diffusion-wave · B-spline · NURBS · Error estimate Mathematics Subject Classification (2010) 26A33 · 74S05 · 65M60 · 65D07

1 Introduction Time-fractional partial differential equations as fractional-order differential problems describe physical models, which have important applications in memory behaviors  Shengfeng Zhu

[email protected] Xindi Hu [email protected] 1

School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

2

Department of Data Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

Numerical Algorithms

of long-time heavy tail decay [22]. The Brownian motion describes normal diffusion in a Gaussian process. As an anomalous diffusion process, subdiffusion is the macroscopic counterpart of continuous time random walk [10]. Diffusion-wave equations [19] can model acoustic and elastic wave propagation dynamics in complex media. For time-dependent partial differential equations [17], the spatial discretization is significant in accuracy and efficiency for fully discrete schemes. A variety of numerical methods are used for spatial discretization of time-fractional partial differential equations. Most of existing spatial discretization methods include finite differences [5, 25, 32], finite elements [9, 12, 13, 20, 28–31], spectral methods [3, 18], spectral element methods [33], finite volumes [8, 27], etc. Recently, Hughes et al. [4, 11] proposed a novel discretization method—isogeometric analysis (IGA)—for solving partial differential equations. In this paper, we use IGA for spatial discretization of time-fractional partial differential equations. The motivation stems from the fact that the geometrical approximation of the computational domain may affe