Abstract: The existing results in the literatures show that if a $(2K+1)$-point conservative finite-difference scheme for hyperbolic conservation laws is conservative and converges as $\Delta x, \Delta t\rightarrow 0$, they converge to the unique entropy solution in several space dimensions. Due to this good property, first--order accurate monotone schemes have played a very important role in designing modern high resolution shock-capturing schemes.
Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical view of point and show that a (2K+1)-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.
Conservation Laws Preprint Server <conservation@math.ntnu.no> Last modified: Thu May 22 11:28:48 MEST 2003