The limitations of non-monotone numerical methods
It is easy to see why oscillations in numerical solutions are often inconsis tent with the behaviour of the exact solution representing some real physical phenomenon. Consider, for example, the one-dimensional convection-diffusion problem (2.3). The derivatives of its exact solution satisfy the equation
This implies that, at each point x € H, the first and second derivatives are either both zero or of opposite sign. Therefore the exact solution cannot oscillate, because such behaviour requires the existence of points in the domain Q. at which the first and second derivatives are non-zero and have the same sign.