Fractional spectral collocation methods for linear and nonlinear variable order FPDEs

• Published in: Journal of computational physics 2015-07, Vol.293 (C), p.312-338
• Main Authors: Zayernouri, Mohsen, Karniadakis, George Em
• Format: Article
• Language: English
• Subjects: Advection-diffusion equation; Collocation methods; Diffusion; Mathematical analysis; Mathematical models; Nonlinearity; Spectra; Spectral lines
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.12.001

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 eigenproblems, we introduce a new family of interpolants, called left-/right-sided and central fractional Lagrange interpolants. We employ the fractional derivatives of (Ieft-/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 timeand 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.