A Hybrid Reconstruction Method Based upon the Characteristic Fields Decomposition of Compressible Euler Equations (Advances in Applied Mathematics and Mechanics, 2025)
The local characteristic fields of compressible Euler equations can be decomposed into linearly degenerate and genuinely nonlinear, respectively. The former is associated with a contact discontinuity (the slip lines for multi-dimensional cases) and the latter is associated with shocks and rarefaction waves (Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009). The weighted essentially non-oscillatory (WENO) scheme can capture shock waves well but its resolution for short waves still needs to be improved, while one kind of new scheme constructed by the boundary variation diminishing (BVD) principle with the THINC (Tangent of Hyperbola for INterface Capturing) reconstruction can resolve the contact discontinuities with higher resolution but it may generate spurious numerical phenomena in some special cases. Combining the physical characteristics and the advantages of WENO and BVD schemes, this paper proposes a hybrid reconstruction method, in which the classical WENO scheme is used to discretize the genuinely nonlinear fields and the higher resolution BVD scheme is used to discretize the linearly degenerate field(s). Numerical experiments show that the hybrid method can preserve high accuracy near both contact discontinuities and short waves as the used higher resolution scheme while effectively avoiding spurious numerical phenomena (such as oscillations and overshoots). Compared with the BVD scheme, the hybrid method also improves computational efficiency, since it uses a more efficient WENO scheme in genuinely nonlinear fields.