A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection. (11th July 2016) - Sun, Ziyao; Xiao, Feng
Summary: Numerical 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 that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment 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 weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values 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 fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd. Abstract : This paper presents a novel algorithm that introduces a nodal value‐based WENO limiter for CIP/MM FVM as a trial to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, which is free of the ad hoc TVB 'trouble cell' indicator can achieve superior accuracy compared with Eulerian formulation due to its semi‐Lagrangian nature. The numerical results of benchmark tests show excellent solution quality compared with other existing schemes.