Rankine–Hugoniot–Riemann solver for steady multidimensional conservation laws with source terms

•We develop a solver for steady multidimensional conservation laws with source terms.•The fluxes in other dimensions are treated as source terms.•The total source term is imposed in the middle of each cell, causing a jump.•The solver is shown to be of second order in space analytically and numerically.•The accuracy of the solver is better than or comparable to that of a reference MUSCL scheme. The Rankine–Hugoniot–Riemann (RHR) solver has been designed to solve steady multidimensional conservation laws with source terms. The solver uses a novel way of incorporating cross fluxes as source terms. The combined source term from the cross fluxes and normal source terms is imposed in the middle of a cell, causing a jump in the solution according to the Rankine–Hugoniot condition. The resulting Riemann problems at the cell faces are then solved by a conventional Riemann solver. We prove that the solver is of second order accuracy for rectangular grids and confirm this by its application to the 2D scalar advection equation, the 2D isothermal Euler equations and the 2D shallow water equations. For these cases, the error of the RHR solver is comparable to or smaller than that of a standard Riemann solver with a MUSCL scheme. The RHR solver is also applied to the 2D full Euler equations for a channel flow with injection, and shown to be comparable to a MUSCL solver.