A numerical scheme for doubly nonlocal conservation laws

• Published in: Calcolo 2024, Vol.61 (4), Article 72
• Main Authors: Abreu, E., Valencia-Guevara, J. C., Huacasi-Machaca, M., Pérez, J.
• Format: Article
• Language: English
• Subjects: Boundary value problems; Conservation laws; Error analysis; Hilbert transformation; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear dynamics; Nonlinearity; Numerical Analysis; Operators (mathematics); Porous media; Theory of Computation
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-024-00624-x

In this work, we consider the nonlinear dynamics and computational aspects for non-negative solutions of one-dimensional doubly nonlocal fractional conservation laws ∂ t u + ∂ x [ u Λ α - 1 ( κ ( x ) H u ) ] = 0 and ∂ t u - ∂ x [ u Λ α - 1 ( κ ( x ) H u ) ] = 0 , where Λ α - 1 denotes the fractional Riesz transform, H denotes the Hilbert transform, and κ ( x ) denotes the spatial variability of the permeability coefficient in a porous medium. We construct an unconventional Lagrangian–Eulerian scheme, based on the concept of no-flow curves, to handle the doubly nonlocal term, under a weak CFL stability condition , which avoids the computation of the derivative of the nonlocal flux function. Primarily, we develop a feasible computational method and derive error estimates of the approximations of the Riesz potential operator Λ α - 1 . Secondly, we undertake a formal numerical-analytical study of initial value problems associated with such doubly nonlocal models to add insights into the role of nonlinearity and coefficient κ ( x ) in the composition between the Hilbert transform and the fractional Riesz potential. Numerical experiments are presented to show the performance of the approach.