A one-parameter family of explicit and implicit upwind second-order-accurate, total variation diminishing (TVD) schemes has been developed by Harten. These TVD schemes have the property of not generating spurious oscillations when applied to one-dimensional nonlinear scalar hyperbolic conservation laws and constant coeffi-cient hyperbolic systems. The goal of this work is to extend these methods to the multidimensional hyperbolic conservation laws in curvilinear coordinates. Various ways of linearizing the implicit operator and solution strategies to improve the computation efficiency of the implicit algorithm are discussed. Numerical experiments with some AGARD test cases for steady-state airfoil calculations show that the proposed linearized implicit up-wind TVD schemes are quite robust and accurate.
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