Residual distribution schemes for advection-diffusion problems on quadrilateral cells

This paper provides the analysis and development of residual distribution schemes for the scalar advection-diffusion problem on quadrilateral cells. The study is performed using the Fourier and the truncation error analyses, and the results are confirmed by numerical experiments. A generalized modified wavenumber is defined which provides a general framework for the truly multidimensional spectral analysis and comparison of schemes belonging to different classes. The analysis demonstrates that using a hybrid approach, which employs an upwind residual distribution scheme for the advection term and any other scheme for the diffusion term, leads to a first-order-accurate method. On the other hand, distributing the entire residual by an upwind scheme provides second-order accuracy; such an approach is unstable in diffusion dominated problems since residual distribution schemes are characterized by undamped modes associated with the discretization of the diffusion term. In this work, such a problem is addressed by defining the conditions for a stable hybrid approach to be second-order-accurate and an optimal scheme having minimal dispersion error on a nine-point stencil is provided