A new type of high-order multi-resolution trigonometric WENO schemes with adaptive linear weights for hyperbolic conservation laws

Abstract

This article provides a series of high-order multi-resolution trigonometric weighted essentially non-oscillatory schemes with adaptive linear weights for solving hyperbolic conservation laws in a finite difference framework, which are termed as the MR-TWENO-ALW schemes. These new TWENO schemes only use the information defined on two unequal-sized spatial stencils and do not need to introduce other stencils to achieve optimal high-order accuracy. To increase the flexibility of the linear weights, we design an adaptive linear weight process which is an automatic adjustment of two linear weights with two simple conditions. This ensures the schemes to get the optimal order of accuracy in smooth regions, accurately approximate sharp gradients, and suppress high oscillations near strong discontinuities. These new MR-TWENO-ALW schemes can achieve high spectral resolution and maintain low computational cost in large scale engineering applications. And these new schemes are simple in the construction and could be extended to arbitrarily high-order accuracy on other computing meshes. Extensive one-dimensional and two-dimensional numerical examples are used to testify the feasibility of these new MR-TWENO-ALW schemes.