Fractional spectral collocation methods for linear and nonlinear variable order FPDEs

JOURNAL OF COMPUTATIONAL PHYSICS

卷:293

Article

Zayernouri, Mohsen¹  Karniadakis, George Em¹ 

[1] Brown Univ, Div Appl Math, Providence, RI 02912 USA

关键词: Jacobi polyfractonomial;    Space-time fractional Lagrange interpolants;    Variable-order differentiation matrix;    Riemann-Liouville;    Riesz and Caputo fractional derivatives;    Penalty method;   

DOI  :  10.1016/j.jcp.2014.12.001

来源: Elsevier

【 摘 要 】

While several high-order methods have been developed for fractional PDEs (FPDEs) with fixed order, there are no such methods for FPDEs with field-variable order. These equations allow multiphysics simulations seamlessly, e.g. from diffusion to sub-diffusion or from wave dynamics transitioning to diffusion, by simply varying the fractional order as a function of space or time. We develop an exponentially accurate fractional spectral collocation method for solving linear/nonlinear FPDEs with field-variable order. Following the spectral theory, developed in [1] for fractional Sturm-Liouville eigen problems, we introduce a new family of interpolants, called left-/right-sided and central fractional Lagrange interpolants. We employ the fractional derivatives of (left-/right-sided) Riemann-Liouville and Riesz type and obtain the corresponding fractional differentiation matrices by collocating the field-variable fractional orders. We solve several FPDEs including time and space-fractional advection-equation, time- and space-fractional advection-diffusion equation, and finally the space-fractional Burgers' equation to demonstrate the performance of the method. In addition, we develop a spectral penalty method for enforcing inhomogeneous initial conditions. Our numerical results confirm the exponential-like convergence of the proposed fractional collocation methods. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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