A Group of CFL-Dependent Flux-Limiters to Control the Numerical Dissipation in Multi-stage Unsteady Calculation

• Published in: Journal of scientific computing Vol. 81; no. 1; pp. 186 - 216
• Main Authors: Ren, Jiong, Wang, Gang, Ma, Mingsheng
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2019; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Costs; Criteria; Design; Entropy; Euler-Lagrange equation; Finite volume method; High resolution; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Numerical dissipation; Parameters; Theoretical; Variables; TVD criterion; High resolution; CFL-dependent flux-limiter; Numerical dissipation
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-019-01010-4

In this paper, a group of CFL-dependent Piecewise-Linear flux-limiters termed as GPL-CFL limiters are proposed to control and limit the numerical dissipation in Multi-stage Unsteady high resolution schemes. These limiters satisfy the second-order TVD criterion while are subject to the Multi-stage Time-space separated Unsteady TVD criterion. The accuracy of the high resolution schemes using these limiters can be improved by properly tuning the four parameters including CFL number and other three variables contained in them. To fully illustrate the ability of GPL-CFL limiters to control dissipation, an entropy dissipation function with these limiters is defined and is proved to be monotonic with each parameter. For some endpoint values of parameters, a relatively ideal limiter named by MDF-CFL limiter is obtained in the sense of CFL-dependent minimum dissipation. In the end, the high resolution entropy-consistent schemes which are termed as HRECSIS schemes are constructed using GPL-CFL limiters and are used to compute the solutions for Burgers and Euler equations in 1D and 2D cases. Numerical results sufficiently illustrate the accuracy of HRECSIS schemes, in the context of MTU calculations, are closely related with the four parameters contained in GPL-CFL limiters and the higher resolution than their corresponding CFL-independent flux-limiters can be obtained for computed solutions when CFL number is in the range of (0, 0.5).