The QUICK scheme is a third‐order finite‐volume scheme with point‐valued numerical solutions

Abstract

In this paper, we resolve the ever-present confusion over the Quadratic Upwind Interpolation for Convective Kinematics (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 economical high-order unstructured-grid schemes as we will discuss in a subsequent paper.

Resolving Confusion Over Third-Order Accuracy of U-MUSCL

“…[4], Burg performed an accuracy verification test for a one-dimensional steady problem with a cell-averaged source term and apparently with point-valued solutions. Therefore, his scheme corresponds to the QUICK scheme [18]. This is the reason that he obtained third-order accuracy with κ = 1/2, not κ = 1/3.…”

“…Other strategies are available for expressing the time derivative terms with point-valued solutions as discussed in Ref. [18]. An example is shown in Ref.…”

“…Because of the switch to the point-valued solution, we obtain a slightly different truncation error, as derived in Ref. [18],…”

“…We consider the finite-difference-type U-MUSCL scheme (3.16) in Section 3.4. This scheme cannot be thirdorder accurate for nonlinear equations because of the second-order error generated from the averaged flux term in the numerical flux (3.3), not the dissipation [12,15,18]. To eliminate this error, we need to reconstruct the flux directly, e.g., by applying the U-MUSCL flux reconstruction with a parameter θ:…”

“…Consider the typical U-MUSCL scheme (3.16) and its truncation error (3.17), it is noted that κ = 1/2 reconstructs a quadratic solution exactly at a face from point-valued solutions stored at nodes [18] (see also the next section). Then, the U-MUSCL scheme (3.16) is essentially equivalent to the second-order central finite-difference scheme; indeed, for κ = 1/2, the truncation error (3.17) becomes…”