Communications on Applied Mathematics and Computation >

2020 , Vol. 2 >Issue 3: 429 - 460

DOI: https://doi.org/10.1007/s42967-019-00044-7

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

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  • 1 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China;
    2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Received date: 2019-05-02

  Revised date: 2019-08-09

  Online published: 2020-05-12

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Abstract

A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659–683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202–228, 1996) might sufer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order fnite-diference and fnite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain ffth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to frst-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one. This is the frst time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order fnitediference and fnite-volume WENO schemes for solving steady-state problems. In comparison with the classical ffth-order fnite-diference and fnite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

Key words： High-order multi-resolution WENO scheme; Unequal-sized hierarchical stencil; Central spatial stencil; Steady-state problem

Cite this article

Jun Zhu, Chi, Wang Shu . Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study[J]. Communications on Applied Mathematics and Computation, 2020 , 2(3) : 429 -460 . DOI: 10.1007/s42967-019-00044-7

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