A new type of non-polynomial based TENO scheme for hyperbolic conservation laws

For classical WENO/TENO reconstructions, the high-order polynomial interpolation is one of the main building-blocks for constructing the numerical fluxes. However, due to the inherent characteristics of polynomials, the polynomial interpolation is not suitable to represent the flow scale with sharp transitions such as high-wavenumber fluctuations or discontinuities. In this work, based on a new candidate stencil arrangement with explicit detection of non-smooth regions, a new shock-capturing framework is proposed by combining the infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with the jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme not only achieves the desired high order accuracy for smooth flow scales but also resolves the genuine discontinuities with a sub-cell resolution. Moreover, with the observation that the built-in parameter has a significant impact on the basic shape of the Gaussian RBF kernel function, a nonlinear adaptation strategy for the shape parameter is proposed to incorporate the local features of the flow field and further minimize the numerical dissipation errors for smooth flow scales. In order to demonstrate the robust shock capturing and high wave-resolution properties of the proposed non-polynomial based scheme, a set of challenging benchmark cases with a wide range of Mach numbers is simulated without the necessity of parameter tuning case by case.

Automated summary

In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.