A simple, robust and efficient high-order accurate shock-capturing scheme for compressible flows: Towards minimalism

Highlights: • A high order accurate shock-capturing scheme for gas-dynamic equations is developed. • Its performance is largely comparable to the 5th order accurate WENO-Rusanov scheme. • Very high resistance against shock anomalies such as carbuncle phenomenon is achieved. • Its computational cost is 30–40% less than that of the above WENO scheme. • Its materials are elementary or generic, e.g. MUSCL and Lagrange's polynomial. Developed is a high-order accurate shock-capturing scheme for the compressible Euler/Navier–Stokes equations; the formal accuracy is 5th order in space and 4th order in time. The performance and efficiency of the scheme are validated in various numerical tests. The main ingredients of the scheme are nothing special; they are variants of the standard numerical flux, MUSCL, the usual Lagrange's polynomial and the conventional Runge–Kutta method. The scheme can compute a boundary layer accurately with a rational resolution and capture a stationary contact discontinuity sharply without inner points. And yet it is endowed with high resistance against shock anomalies (carbuncle phenomenon, post-shock oscillations, etc.). A good balance between high robustness and low dissipation is achieved by blending three types of numerical fluxes according to physical situation in an intuitively easy-to-understand way. The performance of the scheme is largely comparable to that of WENO5-Rusanov, while its computational cost is 30–40% less than of that of the advanced scheme.