Fast stable finite difference schemes for nonlinear cross-diffusion. (1st November 2021)

• Record Type: Journal Article
• Authors: Lobo, Diogo
• Is Part Of: Computers & mathematics with applications. Volume 101(2021)
• Page Start: 23
• Page End: 37
• Publication Date: 2021-11-01
• Subjects: Nonlinear cross-diffusion -- Fast numerical schemes -- Operator splitting -- Matrix factorization -- Image processing; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2021.06.011
• Languages: English
• ISSNs: 0898-1221

Abstract

Highlights: The dynamics of cross-diffusion leads to a high complexity on implicit finite difference schemes implementations. Two stable operator splitting schemes are proposed to lower the complexity. A stable factorization of the system matrix is derived to fasten the computations. We achieve competitive running times (similar to explicit approaches) for these implicit implementations. Abstract: The dynamics of cross-diffusion models leads to a high computational complexity for implicit difference schemes, turning them unsuitable for tasks that require results in real-time. We propose the use of two operator splitting schemes for nonlinear cross-diffusion processes in order to lower the computational load, and establish their stability properties using discrete L 2 energy methods. Furthermore, by attaining a stable factorization of the system matrix as a forward-backward pass, corresponding to the Thomas algorithm for self-diffusion processes, we show that the use of implicit cross-diffusion can be competitive in terms of execution time, widening the range of viable cross-diffusion coefficients for on-the-fly applications.