The QUICK Scheme is a Third-Order Finite-Volume Scheme with Point-Valued Numerical Solutions

TL;DR

This paper clarifies that the QUICK scheme is a third-order finite-volume method for nonlinear conservation laws with point-valued solutions, resolving previous confusion about its order of accuracy through detailed analysis and numerical tests.

Contribution

It provides a rigorous proof of third-order accuracy for the QUICK scheme and discusses discretization techniques to preserve this accuracy in unsteady problems.

Findings

QUICK scheme is third-order accurate for nonlinear conservation laws.

Careful spatial discretization of time derivatives is essential for unsteady problems.

Numerical tests confirm third-order convergence.

Abstract

In this paper, we resolve the ever-present confusion over the QUICK scheme: it is a second-order scheme or a third-order scheme. The QUICK scheme, as proposed in the original reference [B. P. Leonard, Comput. Methods. Appl. Mech. Eng., 19, (1979), 59-98], is a third-order (not second-order) finite-volume scheme for the integral form of a general nonlinear conservation law with point-valued solutions stored at cell centers as numerical solutions. Third-order accuracy is proved by a careful and detailed truncation error analysis and demonstrated by a series of thorough numerical tests. The QUICK scheme requires a careful spatial discretization of a time derivative to preserve third-order accuracy for unsteady problems. Two techniques are discussed, including the QUICKEST scheme of Leonard. Discussions are given on how the QUICK scheme is mistakenly found to be second-order accurate. This paper is intended to serve as a reference to clarify any confusion about third-order accuracy of the QUICK scheme and also as the basis for clarifying third-order unstructured-grid schemes as we will discuss in a subsequent paper.