An Improvement of Third Order WENO Scheme for Convergence Rate at Critical Points with New Non-linear Weights
- PDF / 2,076,297 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 96 Downloads / 160 Views
An Improvement of Third Order WENO Scheme for Convergence Rate at Critical Points with New Non-linear Weights Anurag Kumar1 · Bhavneet Kaur2 © Foundation for Scientific Research and Technological Innovation 2019
Abstract 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. Keywords Smoothness measurement · WENO · Critical points · Sufficient condition · Convergence analysis · Accuracy
Introduction There are large variations on the solutions of partial differential equations coming up in science and engineering over small portion of the physical domain. A major challenge is to solve such problems within these portions while maintaining the sufficient accuracy in acceptable limits. Huge varieties of productive computational methods have been invented
B
Bhavneet Kaur [email protected] Anurag Kumar [email protected]
1
Department of Mathematics, University of Delhi, Delhi 110007, India
2
Department of Mathematics, Lady Shri Ram College for Women, University of Delhi, Delhi 110024, India
123
Differential Equations and Dynamical Systems
to find the solutions of hyperbolic conservation laws but finite difference and finite volume essentially non-oscillatory (ENO) or weighted essentially non-oscillatory (WENO) schemes are intensively applied in the numerical simulation of compressible flows. Several models of non-linear hyperbolic conservation laws do not have their analytical solutions, if they have, they often contain contact discontinuities, shocks and complex smooth structures. A computational scheme is always better if it provides high order accuracy in smooth regions without producing spurious numerical oscillations essentially near the discontinuities. The computational society has taken many efforts to remove such oscillations near the discontinuities. WENO schemes are very favourable numerical schemes which are used to capture contact discontinuities or shocks. The WENO schemes are also utilized to simulate the difficulties of the advanced areas of mathematical physics such as shallow water
Data Loading...
