In this paper we present a formally fourth-order accurate hybrid-variable
method for the Euler equations in the context of method of lines. The
hybrid-variable (HV) method seeks numerical approximations to both
cell-averages and nodal solutions and evolves them in time simultaneously; and
it is proved in previous work that these methods are inherent superconvergent.
Taking advantage of the superconvergence, the method is built on a third-order
discrete differential operator, which approximates the first spatial derivative
at each grid point, only using the information in the two neighboring cells.
Stability and accuracy analyses are conducted in the one-dimensional case for
the linear advection equation; whereas extension to nonlinear systems including
the Euler equations is achieved using characteristic decomposition and the
incorporation of a residual-consistent viscosity to capture strong
discontinuities. Extensive numerical tests are presented to assess the
numerical performance of the method for both 1D and 2D problems.Comment: 26 page