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Volume 13, Issue 5
A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws

Zhenming Wang, Jun Zhu, Yuchen Yang, Linlin Tian & Ning Zhao

Adv. Appl. Math. Mech., 13 (2021), pp. 1064-1095.

Published online: 2021-06

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  • Abstract

In this paper, an efficient class of finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) schemes with increasingly higher order of accuracy is presented for solving hyperbolic conservation laws on structured meshes. The crucial idea is originated from [44,46]. We only use the information defined on a series of nested unequal-sized central spatial stencils in high-order spatial approximations. These new nested multi-resolution WENO schemes use the same large central stencil as the same order classical WENO schemes [3,21] without increasing the number of the total spatial stencils, could obtain the optimal order of accuracy in smooth regions, and could control spurious oscillations nearby strong shocks or contact discontinuities by gradually degrading from ninth-order to seventh-order, fifth-order, third-order or ultimately to the first-order accuracy. Associated linear weights can be set as any positive numbers on condition that their summation is one. Therefore, these new WENO schemes are simple to construct and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark examples are illustrated to show the good performance of these new nested multi-resolution WENO schemes, especially in terms of robustness and low numerical dissipation.

  • Keywords

Nested multi-resolution scheme, finite difference WENO scheme, unequal-sized central spatial stencil, high-order accuracy, hyperbolic conservation laws.

  • AMS Subject Headings

65M60, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-1064, author = {Zhenming and Wang and and 16658 and and Zhenming Wang and Jun and Zhu and and 16659 and and Jun Zhu and Yuchen and Yang and and 16660 and and Yuchen Yang and Linlin and Tian and and 16661 and and Linlin Tian and Ning and Zhao and and 16662 and and Ning Zhao}, title = {A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {5}, pages = {1064--1095}, abstract = {

In this paper, an efficient class of finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) schemes with increasingly higher order of accuracy is presented for solving hyperbolic conservation laws on structured meshes. The crucial idea is originated from [44,46]. We only use the information defined on a series of nested unequal-sized central spatial stencils in high-order spatial approximations. These new nested multi-resolution WENO schemes use the same large central stencil as the same order classical WENO schemes [3,21] without increasing the number of the total spatial stencils, could obtain the optimal order of accuracy in smooth regions, and could control spurious oscillations nearby strong shocks or contact discontinuities by gradually degrading from ninth-order to seventh-order, fifth-order, third-order or ultimately to the first-order accuracy. Associated linear weights can be set as any positive numbers on condition that their summation is one. Therefore, these new WENO schemes are simple to construct and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark examples are illustrated to show the good performance of these new nested multi-resolution WENO schemes, especially in terms of robustness and low numerical dissipation.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0223}, url = {http://global-sci.org/intro/article_detail/aamm/19254.html} }
TY - JOUR T1 - A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws AU - Wang , Zhenming AU - Zhu , Jun AU - Yang , Yuchen AU - Tian , Linlin AU - Zhao , Ning JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1064 EP - 1095 PY - 2021 DA - 2021/06 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0223 UR - https://global-sci.org/intro/article_detail/aamm/19254.html KW - Nested multi-resolution scheme, finite difference WENO scheme, unequal-sized central spatial stencil, high-order accuracy, hyperbolic conservation laws. AB -

In this paper, an efficient class of finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) schemes with increasingly higher order of accuracy is presented for solving hyperbolic conservation laws on structured meshes. The crucial idea is originated from [44,46]. We only use the information defined on a series of nested unequal-sized central spatial stencils in high-order spatial approximations. These new nested multi-resolution WENO schemes use the same large central stencil as the same order classical WENO schemes [3,21] without increasing the number of the total spatial stencils, could obtain the optimal order of accuracy in smooth regions, and could control spurious oscillations nearby strong shocks or contact discontinuities by gradually degrading from ninth-order to seventh-order, fifth-order, third-order or ultimately to the first-order accuracy. Associated linear weights can be set as any positive numbers on condition that their summation is one. Therefore, these new WENO schemes are simple to construct and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark examples are illustrated to show the good performance of these new nested multi-resolution WENO schemes, especially in terms of robustness and low numerical dissipation.

Zhenming Wang, Jun Zhu, Yuchen Yang, Linlin Tian & Ning Zhao. (1970). A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws. Advances in Applied Mathematics and Mechanics. 13 (5). 1064-1095. doi:10.4208/aamm.OA-2020-0223
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