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

doi:10.1007/s42967-019-00044-7

摘要/Abstract

摘要： 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.

关键词: High-order multi-resolution WENO scheme, Unequal-sized hierarchical stencil, Central spatial stencil, Steady-state problem

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

中图分类号:

65M06
35L65

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.

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