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<p>This paper provides a survey and analyzes the distribution of third-order non-linear weighted essentially non-oscillatory (WENO) weights with respect to the smoothness of the input data and corresponding approximation accuracy of the computed solution of underlying partial differential equations. Existing prevailing non-linear WENO3 weights are detailed, and it is demonstrated that even for smooth data, these constructed weights deviate from the optimal weights, and thus resulting WENO schemes fail to achieve third-order accuracy in such smooth solution regions. Based on the observations, a CFL-informed smooth solution region is introduced to devise third-order enhanced hybrid WENO schemes such that the non-linear WENO3 weights retain the optimal weights in smooth solution region. Numerical results are presented and compared for various benchmark test problems using hybrid-WENO schemes and respective underlying WENO3 schemes. The computational results clearly demonstrate that the hybrid scheme provides enhanced resolution of discontinuities and attains the third-order convergence rate for smooth solutions.</p>

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Smooth solution region based enhanced hybrid third-order WENO schemes

  • 期刊论文

  • 发表日期：

    2025年1月26日

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  • Vikas Kumar Jayswal,
  • Sabana Parvin,
  • Ritesh Kumar Dubey

摘要

This paper provides a survey and analyzes the distribution of third-order non-linear weighted essentially non-oscillatory (WENO) weights with respect to the smoothness of the input data and corresponding approximation accuracy of the computed solution of underlying partial differential equations. Existing prevailing non-linear WENO3 weights are detailed, and it is demonstrated that even for smooth data, these constructed weights deviate from the optimal weights, and thus resulting WENO schemes fail to achieve third-order accuracy in such smooth solution regions. Based on the observations, a CFL-informed smooth solution region is introduced to devise third-order enhanced hybrid WENO schemes such that the non-linear WENO3 weights retain the optimal weights in smooth solution region. Numerical results are presented and compared for various benchmark test problems using hybrid-WENO schemes and respective underlying WENO3 schemes. The computational results clearly demonstrate that the hybrid scheme provides enhanced resolution of discontinuities and attains the third-order convergence rate for smooth solutions.