Towards the ultimate conservative difference scheme I. The quest of monotonicity

Eq. (2) transforms an initially monotonic distribution such that it remains monotonic at later times; in fact, the original distribution is just shifted over a distance axt. A difference scheme for Eq. (2) can not deliver this exact solution, except for integer values of ~ = aAt/Ax, but it may at least be required to produce monotonic results for all stable values of (~. This requirement is called the monotouicity condition; schemes satisfying it will colloquially be called monotonic schemes.