A numerical scheme for doubly nonlocal conservation laws.

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 , <graphic href="10092_2024_624_Article_Equ52.gif"></graphic> 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. [ABSTRACT FROM AUTHOR]