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 - <p style="text-align: justify;">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.</p>