A new type of increasingly high-order multi-resolution trigonometric WENO schemes for hyperbolic conservation laws and highly oscillatory problems

Abstract

In this paper, we investigate designing a new type of high-order finite difference multi-resolution trigonometric weighted essentially non-oscillatory (TWENO) schemes for solving hyperbolic conservation laws and some benchmark highly oscillatory problems. We only use the information defined on a hierarchy of nested central spatial stencils in a trigonometric polynomial reconstruction framework without introducing any equivalent multi-resolution representations. These new finite difference trigonometric WENO schemes use the same large stencils as the classical WENO schemes (Jiang and Wu, 1996; Shu, 2009), could obtain the optimal order of accuracy in smooth regions, and simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such multi-resolution trigonometric WENO schemes can be any positive numbers on condition that their summation is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference trigonometric WENO schemes. These new trigonometric WENO schemes are simple to construct and can be easily implemented to arbitrary high-order accuracy in multi-dimensions. Some benchmark examples including some highly oscillatory problems are given to demonstrate the robustness and good performance of these new trigonometric WENO schemes.