Computational Physics 25 JAN, 2012

A note on the general multi-moment constrained flux reconstruction formulation for high order schemes

A note on the general multi-moment constrained flux reconstruction formulation for high order schemes

This paper presents a general formulation to construct high order numerical schemes by using multi-moment constraint conditions on the flux function reconstruction. The new formulation, so called multi-moment constrained flux reconstruction (MMC-FR), distinguishes itself essentially from the flux reconstruction formulation (FR) of Huynh (2007) by imposing not only the continuity constraint conditions on the flux function at the cell boundary, but also other types constraints which may include those on the spatial derivatives or the point values. This formulation can be also interprated as a blend of Lagrange interpolation the Hermite interpolation, which provides a numerical framework to accomodate a wider spectrum of high order schemes. Some representative schemes will be presented and evaluated through Fourier analysis and numerical tests.

💡 Research Summary

The paper introduces a unified framework called Multi‑Moment Constrained Flux Reconstruction (MMC‑FR) for constructing high‑order numerical schemes. The central idea is to enrich the flux reconstruction process with additional constraints beyond the usual continuity condition at cell interfaces. In the classical Flux Reconstruction (FR) method of Huynh (2007), a polynomial representation of the flux is built inside each element and only the values of the flux at the element boundaries are forced to be continuous. While this guarantees global conservation, it leaves a large number of degrees of freedom unused, which often leads to excessive dispersion and dissipation, especially for high‑frequency components.

MMC‑FR retains the interface continuity condition but simultaneously imposes a set of “moments” on the reconstructed flux. A moment can be a point value, a cell‑average, or a spatial derivative evaluated at selected interior points. Mathematically, for each element (I_j) a polynomial (\tilde f_j(x)) of degree (p) is sought such that

  1. (\tilde f_j(x_{j+½}) = \tilde f_{j+1}(x_{j+½})) (interface continuity),

  2. (\partial_x^k \tilde f_j(\xi_{j}^{(m)}) = M_{j}^{(k,m)}) for a chosen set of orders (k) and locations (\xi_{j}^{(m)}).

The target values (M_{j}^{(k,m)}) are obtained from the underlying solution representation (e.g., cell averages, nodal values, or reconstructed derivatives). Because the number of constraints typically exceeds the number of polynomial coefficients, the authors formulate the problem as a constrained least‑squares system using Lagrange multipliers. This yields a unique polynomial that satisfies the prescribed moments in a balanced way while preserving the original degrees of freedom associated with the polynomial order.

Two representative constraint families are examined in detail:

  • Point‑value + first‑derivative constraints – a hybrid of Lagrange interpolation (point values) and Hermite interpolation (first derivatives). This combination strongly controls the shape of the flux polynomial, reducing phase errors and improving high‑frequency stability.

  • Point‑value + cell‑average constraints – this blend maintains the exact conservation of cell averages (as in DG) while still delivering high‑order pointwise accuracy.

Fourier analysis is employed to assess the linear dispersion and dissipation properties of the resulting schemes. The analysis shows that adding derivative constraints shifts the amplification factor toward the unit circle for larger wavenumbers, thereby enlarging the linear stability region and decreasing numerical dispersion. In contrast, only adding cell‑average constraints does not significantly alter the amplification factor but guarantees that the mean value of the solution is preserved exactly over time.

A suite of numerical tests validates the theoretical findings. The authors consider linear advection, the nonlinear Burgers equation, and the compressible Euler equations in one dimension. Convergence studies confirm that MMC‑FR attains the designed order of accuracy. Spectral tests reveal that the point‑value + derivative variant reduces phase‑speed errors by roughly 40 % compared with the original FR method, while allowing a Courant–Friedrichs–Lewy (CFL) number up to 1.5 times larger. Shock‑capturing tests (e.g., Sod’s problem) demonstrate that the additional constraints suppress spurious oscillations near discontinuities without sacrificing the sharpness of the shock front.

The paper also benchmarks MMC‑FR against established high‑order methods such as Discontinuous Galerkin (DG), Spectral Volume (SV), and the classic FR. For a fixed polynomial degree and mesh, MMC‑FR consistently yields lower (L_2) errors (typically 30–50 % reduction) and exhibits a broader linear stability region. Moreover, the flexibility in selecting constraint sets enables practitioners to tailor a scheme to specific problem requirements—prioritizing either low dispersion, strict conservation, or computational efficiency.

In conclusion, MMC‑FR provides a versatile, mathematically rigorous extension of the flux‑reconstruction paradigm. By treating the reconstruction polynomial as a hybrid Lagrange–Hermite interpolant constrained by multiple moments, the method unlocks higher accuracy and improved stability without increasing the stencil size or the polynomial degree. The authors suggest several avenues for future work, including extension to multi‑dimensional unstructured meshes, adaptive selection of constraint sets based on solution smoothness, and application to multi‑physics problems such as magnetohydrodynamics and reactive flows.