TVD scheme Numerical Instability
 

Hello,

I'm trying to calculate the following equation which is the derivative in 'x' of a distribution function:
d(dxF)/dt = d(Efield.(dvxF))/dx

dxF = Derivate of Distribution Function respect to "x"
Efield = Electric Field
dvxF = Derivative of Distribution Function respect to "velocity in x"

Im using kinetic model that's why I'm using Distributions functions and Phase-space. Anyway, that's the form of the equation I'm having trouble with

The problem comes because the right hand of the equation is solved by using central difference, but there is a zone where there is a discontinuity in the electric field (It suddenly increases and then goes down again, Its intended to be that way). The electric field is not exactly the same on each point (but more or less the same order of magnitude) but there is a zone where I have in increment so the difference between two adjacent grid is so big that creates an instability. Is there any way to smooth the numerical scheme?

= (E[i+1][j]. dvxF[i+1][j]-E[i-1][j].dvxF[i-1][j])/dx

This is more or less how it goes. Imagine there is a point where the difference between E[i+1][j] and E[i-1][j] is so big that, creating a big gradient in the dxF value.

I've been studying the MUSCL and TVD schemes but i don't quite understand well the procedure