2024年4月21日 星期日
一维双曲守恒律方程基于HWENO限制器的间断型Petrov-Galerkin方法
A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter
摘要

间断型Petrov-Galerkin方法 (DPG) 是一种区别于DG方法新型双曲守恒律方程数值求解方法, 其具备高精度与模板紧致的特点。然而为了克服高阶线性格式在大梯度解附近的非物理振荡, DPG方法常常需要结合限制器函数来得到高分辨率的数值解图像。本文尝试将HWENO作为限制器函数与DPG进行结合, 求解双曲守恒律方程的间断初值问题。时间离散采用单步高精度SSP Runge-Kutta方法, 采用新的基于HWENO过程作为RKDPG方法的限制器, 仅需一次重构, 就可完成高阶矩的更新, 且无需计算线性权系数。由于精度达不到设计要求, 因此对上述HWENO限制器对上述HWENO限制器中对问题单元的识别方法进行部分改进, 光滑解处使用原有数值解。本文只给出P1元~P3元的计算结果, 该限制器也适用于更高次元的DPG方法。通过数值算例验证了该限制器在问题单元可以有效抑制非物理振荡, 并且保持非问题单元处的原有精度。满足DPG方法的高精度与紧致性特点。具有良好的数值计算效果。

Abstract

The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.  

DOI10.48014/fcpm.20221019002
文章类型研究性论文
收稿日期2022-10-21
接收日期2022-11-07
出版日期2023-06-28
关键词间断型Petrov-Galerkin方法, 双曲守恒律, HWENO 限制器, Runge-Kutta
KeywordsDiscontinuous Petrov-Galerkin Method, Hyperbolic Conservation Law, Hermite WENO Limiter, Runge-Kutta
作者段晓峰, 高巍*
AuthorDUAN Xiaofeng, GAO Wei*
所在单位内蒙古大学数学科学学院, 呼和浩特 010021
CompanySchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
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下载量133
基金项目内蒙古大学科研发展基金(21100-5187133),
内蒙古自治区人才开发基金项目(12000-1300020240)
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引用本文段晓峰, 高巍. 一维双曲守恒律方程基于HWENO限制器的间断型Petrov-Galerkin方法[J]. 中国理论数学前沿, 2023, 1(1): 1-13.
CitationDUAN Xiaofeng, GAO Wei. A discontinuous Petrov-Galerkin Method for one-dimensional hyperbolic conservation law equations based on HWENO Limiter[J]. Frontiers of Chinese Pure Mathematics, 2023, 1(1): 1-13.