Analysis and numerical approximation of viscosity solutions with shocks: Application to the plasma equation

We consider a new class of Hamilton‐Jacobi equations arising from the convective part of general Fokker‐Planck equations ruled by a non‐negative diffusion function that depends on the unknown and on the gradient of the unknown. The new class of Hamilton‐Jacobi equations represents the propagation of fronts with speed that is a nonlinear function of the signal. The equations contain a nonstandard Hamiltonian that allows the presence of shocks in the solution and these shocks propagate with nonlinear velocity. We focus on the one‐dimensional plasma equation as an example of the general Fokker‐Planck equations having the features we are interested in analyzing. We explore features of the solution of the corresponding Hamilton‐Jacobi plasma equation and propose a suitable fifth order finite difference numerical scheme that approximates the solution in a consistent way with respect to the solution of the associated Fokker‐Planck equation. We present numerical results performed under different initial data with compact support.