A Non-oscillatory Multi-Moment Finite Volume Scheme with Boundary Gradient Switching

Abstract

In this work we propose a new formulation for high-order multi-moment constrained 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 constraint 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 principle 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.

High‐order conservative transport on Yin‐Yang grids using the multi‐moment constrained finite volume method

“…The BGS solver is the same as in Deng et al . (2017) for numerical oscillation suppression, which is effective to deal with discontinuous issues (Gu et al ., 2020). where …”

“…where F i,j and G i,j are fourth-order interpolation polynomials of 𝜆and 𝜙-direction fluxes, respectively, on cell (i, j). The BGS solver is the same as in Deng et al (2017) for numerical oscillation suppression, which is effective to deal with discontinuous issues (Gu et al, 2020).…”

A three‐dimensional positivity‐preserving and conservative multimoment finite‐volume transport model on a cubed‐sphere grid

“…As can be seen from Equations ()–(), the remaining key issue is to determine the boundary derivative of the reconstructed function $$$ {\Phi}_i\left(\xi \right) $$$ in the three‐point MCV scheme. The BGS algorithm (Deng et al, 2017) is adopted to suppress nonphysical numerical oscillations in the high‐order MMFV scheme, as follows.…”

“…These moments can be treated as model variables and are updated separately in time by different approaches. For example, the VIA values in multimoment methods are updated by the flux‐form formulation, while point values can be updated flexibly by either the Eulerian method (Ii and Xiao, 2009; Sun et al, 2015; Deng et al, 2017) or the semi‐Lagrangian method (Xiao et al, 2002; Ii and Xiao, 2007; Sun and Xiao, 2017). The finite‐volume constraint on the cell average assures exact numerical conservation.…”

“…As mentioned above, high-order schemes will produce unphysical oscillations around sharp gradients or discontinuities. To suppress numerical oscillations, an MMFV limiting projection including a convexity-preserving limiter (Xiao and Peng, 2004), an oscillationless limiter based on a rational function (Xiao et al, 2002), a high-order WENO (weighted essentially non-oscillatory) type limiter (Sun et al, 2015), and a high-order boundary gradient switching (BGS) technique (Deng et al, 2017) have been devised.…”