A new approach for a nonlocal, nonlinear conservation law (Q2884638)

The authors develop a new approach to nonlocal nonlinear advection in one space dimension. They postulate a nonlocal advection equation with conventional nonlinear advection \(\psi(u)_x\) replaced by the nonlocal integral operator \(\int \psi((u(y,t)+u(x,t))/2)\phi_o(y-x)dy\), where the kernel \(\phi_o\) is an odd function. A nonlocal variant of viscous regularization is also described. Numerical experiments are presented that compare behavior of the nonlocal Burgers equation to the standard local one.