J Sci Comput (2017) 72:1146–1168
A Non-oscillatory Multi-Moment Finite Volume Scheme
with Boundary Gradient Switching
· Ziyao Sun
· Bin Xie
· Kensuke Yokoi
· Feng Xiao
Received: 6 July 2016 / Revised: 7 February 2017 / Accepted: 11 February 2017 /
Published online: 28 February 2017
© Springer Science+Business Media New York 2017
Abstract In this work we propose a new formulation for high-order multi-moment con-
strained ﬁnite volume (MCV) method. In the one-dimensional building-block scheme, three
local degrees of freedom (DOFs) are equidistantly deﬁned within a grid cell. Two candidate
polynomials for spatial reconstruction of third-order are built by adopting one additional con-
straint condition from the adjacent cells, i.e. the DOF at middle point of left or right neighbour.
A boundary gradient switching (BGS) algorithm based on the variation-minimization princi-
ple is devised to determine the spatial reconstruction from the two candidates, so as to remove
the spurious oscillations around the discontinuities. The resulted non-oscillatory MCV3-BGS
scheme is of fourth-order accuracy and completely free of case-dependent ad hoc parameters.
The widely used benchmark tests of one- and two-dimensional scalar and Euler hyperbolic
conservation laws are solved to verify the performance of the proposed scheme in this paper.
The MCV3-BGS scheme is very promising for the practical applications due to its accuracy,
non-oscillatory feature and algorithmic simplicity.
Keywords Multi-moment method · Finite volume method · Variation-minimization
principle · Non-oscillatory scheme · High-order scheme · Local reconstruction
This work was supported in part by JSPS KAKENHI (Grant No. 15H03916) and National Natural Science
Foundation of China (Grant No. 41522504).
Department of Mechanical Engineering, Tokyo Institute of Technology, 4259 Nagatsuta,
Midori-ku, Yokohama 226-8502, Japan
School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, UK
School of Human Settlement and Civil Engineering & State Key Laboratory for Strength and
Vibration of Mechanical Structures, Xi’an Jiaotong University,
Xi’an, Shaanxi 710049, China