Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems

This paper provides a two-dimensional fluctuation splitting scheme for unsteady hyperbolic problems which achieves third-order accuracy in both space and time. For a scalar conservation law, the sufficient conditions for a stable fluctuation splitting scheme to achieve a prescribed order of accuracy in both space and time are derived. Then, using a quadratic space approximation of the solution over each triangular element, based on the reconstruction of the gradient at the three vertices, and a four-level backward discretization of the time derivative, an implicit third-order-accurate scheme is designed. Such a scheme is extended to the Euler system and is validated versus well-known scalar-advection problems and inviscid discontinuous flows.