Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems

The aim of this paper is to construct upwind residual distribution schemes for the time accurate solution of hyperbolic conservation laws. To do so, we evaluate a space–time fluctuation based on a space–time approximation of the solution and develop new residual distribution schemes which are extensions of classical steady upwind residual distribution schemes. This method has been applied to the solution of scalar advection equation and to the solution of the compressible Euler equations both in two space dimensions. The first version of the scheme is shown to be, at least in its first order version, unconditionally energy stable and possibly conditionally monotonicity preserving. Using an idea of Csik et al. [Space–time residual distribution schemes for hyperbolic conservation laws, 15th AIAA Computational Fluid Dynamics Conference, Anahein, CA, USA, AIAA 2001-2617, June 2001], we modify the formulation to end up with a scheme that is unconditionally energy stable and unconditionally monotonicity preserving. Several numerical examples are shown to demonstrate the stability and accuracy of the method.