Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

• Published in: Journal of computational physics Vol. 468; p. 111513
• Main Authors: Yu, Bing, Zheng, Xiangcheng, Zhang, Pingwen, Zhang, Lei
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.11.2022; Elsevier Science Ltd
• Subjects: Algorithms; Approximation; Computational physics; Fractional Laplacian; Integers; Phase field; Saddle dynamics; Solution landscape; Stationary solution; Variable-order; Stationary solution; Variable-order; Saddle dynamics; Fractional Laplacian; Solution landscape; Phase field
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111513

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems. •A fast approximation algorithm for variable-order spectral fractional Laplacian.•Construction of solution landscapes of space-fractional phase field models.•Reconfiguration of fractional Laplacian problems by integer-order problems.