High-order gas-kinetic scheme with TENO class reconstruction for the Euler and Navier-Stokes equations

The high-order gas-kinetic scheme (HGKS) with WENO spatial reconstruction method has been extensively validated through numerous numerical experiments, demonstrating its superior accuracy, efficiency and robustness. In comparison to WENO class schemes, TENO class schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. This paper introduces two types of fifth-order HGKS utilizing TENO class schemes. One approach involves replacing the WENO scheme with the TENO scheme in the conventional WENO GKS. Both WENO and TENO schemes provide non-equilibrium state values at the cell interface. The slopes of the non-equilibrium state, as well as the values and slopes of the equilibrium state, are obtained through additional linear reconstruction. Another type of HGKS scheme named TENO GKS with adaptive order (TENO-A GKS), is similar to most multi-resolution HGKS schemes. Following a robust scale-separation procedure, a tailored ENO-like stencil selection strategy is proposed to ensure that high-order accuracy is maintained in smooth regions by selecting the candidate reconstruction from a large stencil, while enforcing the ENO property near discontinuities by adopting the candidate reconstruction from small smooth stencils. These TENO schemes include TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) and TENO schemes with dual ENO-like stencil selection (TENO-D). The HGKS scheme based on TENO-A reconstruction takes advantage of the large stencil to provide point values and slopes of the non-equilibrium states. By dynamically merging the reconstructed non-equilibrium slopes, the need for extra reconstruction of the equilibrium states at the beginning of each time step can be avoided. The simple TENO-A GKS scheme exhibits better robustness compared to the conventional WENO GKS or TENO GKS.