Explicit Runge–Kutta residual distribution schemes for time dependent problems: Second order case

In this paper, we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems, starting from a given residual distribution (RD) discrete approximation of the steady operator. We review the existing knowledge on consistent RD mass matrices and highlight the relations between different definitions. We then introduce our explicit approach which is based on three main ingredients: first recast the RD discretization as a stabilized Galerkin scheme, then use a shifted time discretization in the stabilization operator, and lastly apply high order mass lumping on the Galerkin component of the discretization. The discussion is particularly relevant for schemes of the residual distribution type [18] and [3] which we will use for all our numerical experiments. However, similar ideas can be used in the context of residual-based finite volume discretizations such as the ones proposed in [14] and [12]. The schemes are tested on a wide variety of classical problems confirming the theoretical expectations.