An Improvement of Third Order WENO Scheme for Convergence Rate at Critical Points with New Non-linear Weights
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In this paper, we construct and implement a new improvement of third order weighted essentially non-oscillatory (WENO) scheme in the finite difference framework for hyperbolic conservation laws. In our approach, a modification in the global smoothness measurement is reported by applying all three points on global stencil \((i-1,i,i+1)\) which is used for convergence of non-linear weights towards the optimal weights at critical points and achieves the desired order of accuracy for third order WENO scheme. We use the third order accurate total variation diminishing (TVD) Runge-Kutta time stepping method. The major advantage of the proposed scheme is its better numerical accuracy in smooth regions. The computational performance of the proposed WENO scheme with this global smoothness measurement is verified in several benchmark one- and two-dimensional test cases for scalar and vector hyperbolic equations. Extensive computational results confirm that the new proposed scheme achieves better performance as compared with WENO-JS3, WENO-Z3 and WENO-F3 schemes.
KeywordsSmoothness measurement WENO Critical points Sufficient condition Convergence analysis Accuracy
The authors are thankful to Center for Fundamental Research in Space Dynamics and Celestial Mechanics (CFRSC), New Delhi, Delhi, India for providing research facilities. We also express gratitude to CSIR, Govt. of India for the grant reference no. 09/045(1438)/2016-EMR-I.