Economically High-Order Unstructured-Grid Methods: Clarification and Efficient FSR Schemes

TL;DR

This paper clarifies reconstruction-based discretization schemes for unstructured grids and introduces efficient flux-and-solution-reconstruction (FSR) methods that achieve high-order accuracy with minimal extra cost, verified through numerical experiments.

Contribution

It introduces and clarifies economical high-order FSR schemes for unstructured grids, combining extended kappa-scheme with flux reconstruction for efficient high-order accuracy.

Findings

Achieves up to fifth-order accuracy on unstructured grids.

Demonstrates formal order of accuracy through numerical experiments.

Provides a set of economical FSR schemes for practical use.

Abstract

In this paper, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended kappa-scheme combined with economical flux reconstruction formulas, achieving up to fifth-order accuracy (sixth-order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.