Residual distribution schemes for advection and advection-diffusion problems on quadrilateral cells

This paper provides a study of some difficulties arising when extending residual distribution schemes for scalar advection and advection-diffusion problems from triangular grids to quadrilateral ones. The Fourier and truncation error analyses on a structured mesh are employed and a generalized modified wavenumber is defined, which provides a general framework for the multidimensional analysis and comparison of different schemes. It is shown that, for the advection equation, linearity preserving schemes for quadrilaterals provide lower dissipation with respect to their triangle-based counterparts and very low or no damping for high frequency Fourier modes on structured grids; therefore, they require an additional artificial dissipation term for damping marginally stable modes in order to be employed with success for pure advection problems. In the case of advection-diffusion problems, a hybrid approach using an upwind residual distribution scheme for the convective fluctuation and any other scheme for the diffusion term is only first-order accurate. On the other hand, distributing the entire residual by an upwind scheme provides second-order accuracy; however, such an approach is unstable for diffusion dominated problems, since residual distribution schemes are characterized by undamped modes associated with the discretization of the diffusive fluctuation. The present analysis allows one to determine the conditions for a stable hybrid approach to be second-order accurate and to design an optimal scheme having minimum dispersion error on a nine-point stencil. Well-documented test-cases for advection and advection-diffusion problems are used to compare the accuracy properties of several schemes.