A High-Order Compact Finite Volume Method for Unstructured Grids: Scheme Space Formulation and One-Dimensional Implementations

TL;DR

This paper introduces a new high-order compact finite volume scheme for unstructured grids, utilizing a generalized null space approach to construct schemes with controllable dispersion and dissipation, combined with WENO for shock capturing.

Contribution

A novel scheme space formulation transforms high-order compact scheme construction into solving null spaces, enabling flexible dispersion control and improved shock-capturing capabilities.

Findings

Achieves high-order accuracy and robustness.

Effectively captures strong discontinuities.

Provides detailed Fourier analysis of scheme properties.

Abstract

This paper presents a novel and straightforward compact reconstruction procedure for the high-order finite volume method on unstructured grids. In this procedure, we constructed a linear approximation relationship between the mean values and the function values, as well as the derivative values. Compared with the classical compact schemes, which employ a Taylor expansion method to determine the coefficients, our approach adopts an equivalent and more generalized method to achieve this goal. Via this method, the problem of constructing a high-order compact scheme is transformed into solving the null space of undetermined homogeneous linear systems. This null space constitutes the complete set of schemes that meet the specified accuracy under a given stencil, and is termed the 'scheme space'. Schemes within the scheme space possess the same accuracy level yet exhibit distinct dispersion and dissipation characteristics. Through Fourier analysis, we can get the dissipation and dispersion properties of all schemes in the scheme space. This facilitates the control of scheme dispersion and dissipation without altering the stencil compactness. Combined with the WENO (Weighted Essentially Non-Oscillatory) concept, multi-stencil schemes are employed to construct the nonlinear weighted compact finite volume scheme (WCFV). The WCFV is capable of eliminating unphysical oscillations at discontinuities, thereby enabling the capture of strong discontinuities. One-dimensional schemes are discussed in detail, and numerical results demonstrate that the proposed method exhibits high-order accuracy, robustness, and shock-capturing capability.