Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 293, Issue -, Pages 312-338Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2014.12.001
Keywords
Jacobi polyfractonomial; Space-time fractional Lagrange interpolants; Variable-order differentiation matrix; Riemann-Liouville; Riesz and Caputo fractional derivatives; Penalty method
Funding
- Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4) at PNNL - U.S. Department of Energy
- AFOSR MURI
- NSF/DMS
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1216437] Funding Source: National Science Foundation
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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.
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