A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Qianqian Yang, Ian Turner, Timothy Moroney, Fawang Liu
2014 Applied Mathematical Modelling  
A finite volume scheme with preconditioned Lanczos method for twodimensional space-fractional reaction-diffusion equations. Abstract Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of
more » ... tional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction-diffusion equations. The computational heart of this approach is the efficient computation of a matrixfunction-vector product f (A)b, where A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is showcased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.
doi:10.1016/j.apm.2014.02.005 fatcat:dydqjt2i5bamrp4npago4kgccu