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.
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