A Numerical Approach to Scalar Nonlocal Conservation Laws
This work provides numerical tools and insights for mathematicians studying nonlocal conservation laws, but the results are primarily numerical and conjectural.
The paper develops a convergent numerical algorithm for 1D scalar nonlocal conservation laws and uses it to show that standard properties of local conservation laws fail in the nonlocal setting, while also conjecturing convergence of nonlocal to local equations based on numerical evidence.
We address the study of a class of 1D nonlocal conservation laws from a numerical point of view. First, we present an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are lead to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.