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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

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. 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

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# Aerospace Hyperbolic Equation Instability

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