In this work, we consider the nonlinear dynamics and computational aspects for non-negative solutions of one-dimensional doubly nonlocal fractional conservation laws \(\begin{aligned} \partial _t u+\partial _x[u\Lambda ^{\alpha -1}(\kappa (x)\mathcal {H}u)]=0 \quad \text { and } \quad \partial _t u-\partial _x[u\Lambda ^{\alpha -1}(\kappa (x)\mathcal {H}u)]=0, \end{aligned}\) where \(\Lambda ^{\alpha -1}\) denotes the fractional Riesz transform, \(\mathcal {H}\) denotes the Hilbert transform, and \(\kappa (x)\) 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 \(\Lambda ^{\alpha -1}\) . 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 \(\kappa (x)\) in the composition between the Hilbert transform and the fractional Riesz potential. Numerical experiments are presented to show the performance of the approach.
