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Volume 36, Issue 1
High Order Compact Hermite Reconstructions and Their Application in the Improved Two-Stage Fourth Order Time-Stepping Framework for Hyperbolic Problems: Two-Dimensional Case

Ang Li, Jiequan Li, Juan Cheng & Chi-Wang Shu

Commun. Comput. Phys., 36 (2024), pp. 1-29.

Published online: 2024-07

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  • Abstract

The accuracy and efficiency of numerical methods are hot topics in computational fluid dynamics. In the previous work [J. Comput. Phys. 355 (2018) 385] of Du et al., a two-stage fourth order ($S_2O_4$) numerical scheme for hyperbolic conservation laws is proposed, which is based on dimension-by-dimensional HWENO5 and WENO5 reconstructions and GRP solver, and uses a $S_2O_4$ time-stepping framework. In this paper, we aim to design a new type of $S_2O_4$ finite volume scheme, to further improve the compactness and efficiency of the numerical scheme. We design an improved $S_2O_4$ framework for two-dimensional compressible Euler equations, and develop nonlinear compact Hermite reconstructions to avoid oscillations near discontinuities. The new two-stage fourth order numerical schemes based on the above nonlinear compact Hermite reconstructions and GRP solver are high-order, stable, compact, efficient and essentially non-oscillatory. In addition, the above reconstructions and the corresponding numerical schemes are extended to eighth-order accuracy in space, and can be theoretically extended to any even-order accuracy. Finally, we present a large number of numerical examples to verify the excellent performance of the designed numerical schemes.

  • AMS Subject Headings

65M08, 65M12, 35L65, 76M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-36-1, author = {Li , AngLi , JiequanCheng , Juan and Shu , Chi-Wang}, title = {High Order Compact Hermite Reconstructions and Their Application in the Improved Two-Stage Fourth Order Time-Stepping Framework for Hyperbolic Problems: Two-Dimensional Case}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {1}, pages = {1--29}, abstract = {

The accuracy and efficiency of numerical methods are hot topics in computational fluid dynamics. In the previous work [J. Comput. Phys. 355 (2018) 385] of Du et al., a two-stage fourth order ($S_2O_4$) numerical scheme for hyperbolic conservation laws is proposed, which is based on dimension-by-dimensional HWENO5 and WENO5 reconstructions and GRP solver, and uses a $S_2O_4$ time-stepping framework. In this paper, we aim to design a new type of $S_2O_4$ finite volume scheme, to further improve the compactness and efficiency of the numerical scheme. We design an improved $S_2O_4$ framework for two-dimensional compressible Euler equations, and develop nonlinear compact Hermite reconstructions to avoid oscillations near discontinuities. The new two-stage fourth order numerical schemes based on the above nonlinear compact Hermite reconstructions and GRP solver are high-order, stable, compact, efficient and essentially non-oscillatory. In addition, the above reconstructions and the corresponding numerical schemes are extended to eighth-order accuracy in space, and can be theoretically extended to any even-order accuracy. Finally, we present a large number of numerical examples to verify the excellent performance of the designed numerical schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0023}, url = {http://global-sci.org/intro/article_detail/cicp/23295.html} }
TY - JOUR T1 - High Order Compact Hermite Reconstructions and Their Application in the Improved Two-Stage Fourth Order Time-Stepping Framework for Hyperbolic Problems: Two-Dimensional Case AU - Li , Ang AU - Li , Jiequan AU - Cheng , Juan AU - Shu , Chi-Wang JO - Communications in Computational Physics VL - 1 SP - 1 EP - 29 PY - 2024 DA - 2024/07 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2024-0023 UR - https://global-sci.org/intro/article_detail/cicp/23295.html KW - Hyperbolic conservation laws, two-stage fourth order time-stepping framework, compact Hermite reconstruction, high order, GRP solver. AB -

The accuracy and efficiency of numerical methods are hot topics in computational fluid dynamics. In the previous work [J. Comput. Phys. 355 (2018) 385] of Du et al., a two-stage fourth order ($S_2O_4$) numerical scheme for hyperbolic conservation laws is proposed, which is based on dimension-by-dimensional HWENO5 and WENO5 reconstructions and GRP solver, and uses a $S_2O_4$ time-stepping framework. In this paper, we aim to design a new type of $S_2O_4$ finite volume scheme, to further improve the compactness and efficiency of the numerical scheme. We design an improved $S_2O_4$ framework for two-dimensional compressible Euler equations, and develop nonlinear compact Hermite reconstructions to avoid oscillations near discontinuities. The new two-stage fourth order numerical schemes based on the above nonlinear compact Hermite reconstructions and GRP solver are high-order, stable, compact, efficient and essentially non-oscillatory. In addition, the above reconstructions and the corresponding numerical schemes are extended to eighth-order accuracy in space, and can be theoretically extended to any even-order accuracy. Finally, we present a large number of numerical examples to verify the excellent performance of the designed numerical schemes.

Li , AngLi , JiequanCheng , Juan and Shu , Chi-Wang. (2024). High Order Compact Hermite Reconstructions and Their Application in the Improved Two-Stage Fourth Order Time-Stepping Framework for Hyperbolic Problems: Two-Dimensional Case. Communications in Computational Physics. 36 (1). 1-29. doi:10.4208/cicp.OA-2024-0023
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