A Seventh-order Perturbational Weighted Essentially Non-oscillatory Scheme for Hyperbolic Conservation Laws

Abdalla, Amr H.; Azer, Martina W.; Ramadan, Moutaz

doi:10.21608/ajbas.2023.201613.1150

A Seventh-order Perturbational Weighted Essentially Non-oscillatory Scheme for Hyperbolic Conservation Laws

Document Type : Original Article

Authors

¹ Department of Physics and Engineering Mathematics, Faculty of Engineering, Port Said University, Egypt.

² Department of physics and Engineering Mathematics, Faculty of Engineering, Port Said university, Port Said City.

³ Department of mathematics and computer science, faculty of science, port said university, port said, Egypt.

10.21608/ajbas.2023.201613.1150

Abstract

This study presents a modified seventh-order weighted essentially non-oscillatory (WENO) finite difference scheme based on the numerical perturbation method established in [1]. The perturbed candidate polynomials of the seventh-order WENO scheme are evolved using a perturbational polynomial of the grid spacing, which modifies the polynomial approximation used for the classical WENO7-Z reconstruction on each candidate stencil. Furthermore, it is found that the new weighted scheme constructed with the new perturbed polynomials candidate has necessary and sufficient conditions for seventh-order convergence that are one order lower than those used by Henrick for the classic WENO scheme with seventh-order convergence, as presented in [2]. As a result, even at critical locations, the new seventh-order WENO scheme, which uses the perturbed polynomials and the same weights as the WENO7-Z scheme as demonstrated in [3], is able to satisfy the necessary and sufficient condition for seventh-order convergence.

The new WENO7-P scheme reduces numerical dissipation in WENO schemes. Numerical examples verify the new scheme's accuracy, low dissipation, and robustness.

Keywords