Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space

TL;DR

This paper introduces a fifth-order WENO scheme using exponential polynomial approximation with an optimized tension parameter, achieving sixth-order accuracy and improved resolution with reduced numerical dissipation.

Contribution

It develops a novel WENO scheme based on exponential approximation space and introduces a practical method to optimize the tension parameter for enhanced accuracy.

Findings

Achieves sixth-order accuracy without loss at critical points

Reduces numerical dissipation significantly

Demonstrates improved resolution in benchmark tests

Abstract

The aim of this study is to develop a novel WENO scheme that improves the performance of the well-known fifth-order WENO methods. The approximation space consists of exponential polynomials with a tension parameter that may be optimized to fit the the specific feature of the data, yielding better results compared to the polynomial approximation space. However, finding an optimal tension parameter is a very important and difficult problem, indeed a topic of active research. In this regard, this study introduces a practical approach to determine an optimal tension parameter by taking into account the relationship between the tension parameter and the accuracy of the exponential polynomial interpolation under the setting of the fifth-order WENO scheme. As a result, the proposed WENO scheme attains an improved order of accuracy (that is, sixth-order) better than other fifth-order WENO methods without loss of accuracy at critical points. A detailed analysis is provided to verify the improved convergence rate. Further, we present modified nonlinear weights based on $L^1$-norm approach along with a new global smoothness indicator. The proposed nonlinear weights reduce numerical dissipation significantly, while attaining better resolution in smooth regions. Some experimental results for various benchmark test problems are presented to demonstrate the ability of the new scheme.