Multidimensional high-resolution Schemes for viscous hypersonic flows

The first-order multidimensional flux difference splitting method (Parent, B., "Multidimensional Flux Difference Splitting Schemes," AIAA Journal, Vol. 53, No. 7, 2015, pp. 1936-1948) is here extended to second-order accuracy through the use of limiters. The proposed second-order flux functions are such that they collapse to the first-order multidimensional flux difference splitting stencil in the vicinity of discontinuities and tend toward second-order finite difference stencils in smooth parts of the solution. Various test cases of interest to hypersonic flight including supersonic ramp injectors, shock-induced boundary-layer separation, and hypersonic viscous layers reveal that the proposed method achieves as much as a four times increase in resolution per dimension over its first-order counterpart or over the second-order symmetric total variation diminishing schemes. When integrated through a block-implicit approach, this entails as much as a 40-fold increase in computational efficiency while not compromising the symmetric total variation diminishing reliable convergence to steady-state, essentially monotone solution, and high resolution of boundary layers.