A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Abstract

SUMMARYNumerical oscillation has been an open problem for high order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods, and thus are not able to make full use of the sub-cell information available in the local high order reconstructions. This paper presents a novel algorithm which introduces a nodal-value based weighted essentially non-oscillatory limiter for constrained interpolation profile/multimoment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222(2007), 849-871 ) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP-CSL-WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the WENO reconstruction which makes use of the subcell information available from the local DOFs and is built from the point values (PVs) at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the 4th order accuracy and oscillation-suppressing property of the proposed scheme.

Boundary variation diminished conservative semi-Lagrangian method for both compressible and incompressible flows

“…Recently, a WENO-limiter devised for the CIP-CSL scheme (CIP-CSL-WENO4) was proposed by Sun et al [33].…”

“…In the CIP-CSL3-ENO scheme, an ENO indicator is designed, which intentionally selects the non-smooth stencil but can efficiently minimize numerical oscillations. Moreover, Sun et.al [33] proposed a weighted essentially non-oscillatory (WENO) limiter for the CIP-CSL scheme (CIP-CSL-WENO4) to achieve fourth-order accuracy for the smooth solution and keep the WENO property [34,35] for discontinuities. The CIP-CSL-WENO4 scheme manifests the fourth-order accuracy and oscillation-suppressing property for both scalar and Euler conservation laws.…”

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

“…At a certain time point, t, the fluid properties at a specific location, i.e. Sun et al [20] developed a high-order semi-Lagrangian scheme based on finite volume method and a non-oscillatory limiter to control the numerical oscillation, an open problem for high-order numerical methods. the opposite direction of the flow) along the characteristic curve to find the departure point, at which the fluid properties of the previous time moment, t À Δt, are considered the same as those of time t at the arrival point.…”

“…poor conservation of advected properties and/or high numerical dissipations [7,17]. During the past decades, many researchers have tried to solve the problems by developing better discretization schemes [18,19], high-order interpolation schemes [20,21], and hybrid methods combining interpolation schemes with different orders [7]. Zerroukat [17] proposed a semi-Lagrangian method equipped with conservative remapping scheme.…”

“…Zerroukat [17] proposed a semi-Lagrangian method equipped with conservative remapping scheme. Sun et al [20] developed a high-order semi-Lagrangian scheme based on finite volume method and a non-oscillatory limiter to control the numerical oscillation, an open problem for high-order numerical methods. Xiu et al [18] developed higher-order temporal and spatial discretization schemes.…”