High-order multi-moment finite volume method with smoothness adaptive fitting reconstruction for compressible viscous flow

Abstract

We proposed a novel multi-moment finite volume method (MMFVM) to compute compressible viscous flows on arbitrary unstructured grids. Unlike the previous volume-average/point-value multi-moment (VPM) method (Xie et al. 2014 [60]), the present method, so-called VPM-Ex (VPM with extended-stencil reconstruction), formulates novel spatial reconstructions with substantially improved robustness and accuracy for both smooth and discontinuous solutions. Our efforts in this work have brought at least the following new features to the numerical formulation: 1) a more robust and accurate spatial reconstruction based on the stencil of all immediately adjacent cells that supports higher-order limiting projection; 2) two different options for both explicit and implicit reconstructions; 3) a simplified and more accurate formulation for point-value. In addition, we have proposed a smoothness-adaptive fitting (SAF) method for limiting projection to improve the solution quality for discontinuities. The resulting reconstruction method that combines VPM-Ex scheme with SAF-based limiting projection schemes is able to attain 3rd-order accuracy for smooth solution, and non-oscillatory and less-dissipative numerical results for discontinuous solution on unstructured grids of different types of elements. The numerical method has been extensively verified by various benchmark tests for scalar conservation equation, Euler equations and Navier–Stokes equations in 2 and 3 dimensions. The numerical results substantiate that the present method provides a novel numerical formulation of great practical significance to accurately resolve both continuous and discontinuous solutions.