Improved second-order hyperbolic residual-distribution scheme and its extension to third-order on arbitrary triangular grids

In this paper, we construct second- and third-order hyperbolic residual-distribution schemes for general advection–diffusion problems on arbitrary triangular grids. We demonstrate that the accuracy of the second-order hyperbolic schemes in [J. Comput. Phys. 227 (2007) 315–352] and [J. Comput. Phys. 229 (2010) 3989–4016] can be greatly improved by requiring the scheme to preserve exact quadratic solutions. The improved second-order scheme can be easily extended to a third-order scheme by further requiring the exactness for cubic solutions. These schemes are constructed based on the SUPG methodology formulated in the framework of the residual-distribution method, and thus can be considered as economical and powerful alternatives to high-order finite-element methods. For both second- and third-order schemes, we construct a fully implicit solver by the exact residual Jacobian of the proposed second-order scheme, and demonstrate rapid convergence, typically with no more than 10–15 Newton iterations (and about 200–800 linear relaxations per Newton iteration), to reduce the residuals by ten orders of magnitude. We also demonstrate that these schemes can be constructed based on a separate treatment of the advective and diffusive terms, which paves the way for the construction of hyperbolic residual-distribution schemes for the compressible Navier–Stokes equations. Numerical results show that these schemes produce exceptionally accurate and smooth solution gradients on highly skewed and anisotropic triangular grids even for a curved boundary problem, without introducing curved elements. A quadratic reconstruction of the curved boundary normals and a high-order integration technique on curved boundaries are also provided in details.