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The conservation form of the second-order Crowley advection scheme, as some authors have pointed out, may lead to numerical instability for multi-dimensional blows. Replacing the original second-order-accurate approximation of the first spatial partial derivative, with one that includes information about the dimensionality of the field, and also considering the cross-space difference, eliminates this instability without destroying the level of conservation or the accuracy of the original Crowley scheme. The paper also presents some flux correction solutions, the use of which avoids the development of negative values in the solution for positive definite scalars. The paper also discusses the solution of the advection equation in the em of strong deformational flow.
Piotr K. Smolarkiewicz (Wed,) studied this question.