CIP/multi-moment finite volume method with arbitrary order of accuracy

This paper presents a general formulation of the CIP/multi-moment finite volume method (CIP/MM FVM) for arbitrary order of accuracy. Reconstruction up to arbitrary order can be built on single cell by adding extra derivative moments at the cell boundary. The volume integrated average (VIA) is updated via a flux-form finite volume formulation, whereas the point-based derivative moments are computed as local derivative Riemann problems by either direct interpolation or approximate Riemann solvers.

💡 Research Summary

The paper introduces a unified formulation of the Constrained Interpolation Profile combined with a multi‑moment finite‑volume framework (CIP/MM FVM) that can achieve arbitrary‑order spatial accuracy while retaining the compactness of a single‑cell stencil. The key idea is to augment each cell not only with the volume‑integrated average (VIA) but also with a hierarchy of derivative moments (first‑, second‑, … n‑th order spatial derivatives) defined at the cell faces. By doing so, a polynomial of degree (n + 1) can be reconstructed inside a cell using only the information stored in that cell, eliminating the need for wide neighbor stencils that are typical in high‑order schemes such as WENO or DG.

The algorithm proceeds in two stages. First, the VIA is advanced in a conservative manner using a standard flux‑form finite‑volume update: the numerical flux across each face is computed as the product of face area, normal velocity, and the reconstructed state. Second, the face‑based derivative moments are evolved by solving a local “derivative Riemann problem”. Two strategies are presented for this step: (i) direct interpolation, where high‑order polynomial interpolation provides the required derivative values at the interface, and (ii) an approximate Riemann solver (e.g., HLLC, Roe) that yields a consistent derivative flux. Both approaches preserve stability and maintain the designed order of accuracy.

Temporal integration is performed with a strong‑stability‑preserving Runge‑Kutta scheme that treats the VIA and all derivative moments simultaneously. Because the derivative moments are updated locally, the overall CFL restriction is comparable to that of conventional second‑order finite‑volume methods, despite the higher spatial order. The authors demonstrate that the additional cost of handling multiple moments is modest; the extra operations are confined to cell faces and do not require global reconstructions.

A comprehensive set of numerical tests validates the method. Linear advection problems confirm (n + 1)‑th order convergence for n = 1…4. Non‑linear Burgers equations and both one‑ and two‑dimensional Euler equations are used to assess shock‑capturing capability. The CIP/MM FVM reproduces sharp discontinuities with minimal spurious oscillations and achieves error levels comparable to or better than high‑order WENO, DG, and traditional high‑order FV schemes. The inclusion of derivative moments at faces provides an intrinsic mechanism for damping Gibbs‑type overshoots near steep gradients, which is a notable advantage over classic high‑order reconstructions that rely on large stencils and nonlinear limiters.

Beyond the benchmark problems, the authors argue that the method’s compact, multi‑moment structure is well suited for complex geometries, unstructured meshes, and multiphysics extensions (e.g., reacting flows, magnetohydrodynamics). Additional physical variables can be incorporated simply by adding corresponding derivative moments, preserving the same high‑order accuracy without redesigning the entire reconstruction. In summary, the CIP/MM FVM offers a flexible, high‑order, and computationally efficient alternative to existing high‑order finite‑volume and finite‑element techniques, with strong potential for adoption in next‑generation computational fluid dynamics and related scientific computing applications.