High-order shock-capturing hyperbolic residual-distribution schemes on irregular triangular grids

In this paper, we construct second- and third-order non-oscillatory shock-capturing hyperbolic residual-distribution schemes for irregular triangular grids, extending the schemes developed in J. Comput. Phys., 300 (2015), 455-491 to discontinuous problems. We present extended first-order N- and Rusanov-scheme formulations for a hyperbolic advection-diffusion system, and demonstrate that the hyperbolic diffusion term does not have any adverse effect on the solution of inviscid problems for a vanishingly small viscous coefficient. We then construct second- and third-order non-oscillatory hyperbolic residual-distribution schemes by blending the non-monotone second- and third-order schemes with the extended first-order schemes as typically done in the residual-distribution schemes, and examine them for discontinuous problems on irregular triangular grids. We also propose to use the Rusanov scheme to avoid non-physical shocks in combination with an improved characteristics-based nonlinear wave sensor for detecting shocks, compression, and expansion regions. We then verify the design order of accuracy of these blended schemes on irregular triangular grids.