New limiter regions for multidimensional flows

TL;DR

This work tackles the challenge of achieving high-order accuracy in multidimensional incompressible advection while preserving monotonicity and a discrete maximum principle. It shows that the Spekreijse limiter region, though powerful in flux-splitting contexts, is not universally applicable to flux-form incompressible advection; the authors derive two new limiter regions, including Woodfield-type and a differentiable limiter, that guarantee positive-coefficient representations and LED behavior. The paper also analyzes temporal discretisation, linear invariance, and accuracy, and provides extensive numerical demonstrations demonstrating positivity and improved accuracy for the new limiters while preserving monotonicity under MVTS flows. The results offer a practical framework for designing limiter functions that maintain stability and accuracy in multidimensional incompressible transport, with potential impact on atmospheric and oceanic advection modeling and unstructured mesh schemes.

Abstract

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms. The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection.

Figures (12)

• Figure 1.1: \ref{['fig:sweby']} is the plot of the sufficient admissible limiter region in the $(r,y)$ plane as defined by Sweby S_1984. \ref{['fig:spek']} is the plot of the admissible limiter region in the $(R,y)$ plane as defined by Spekreijse S_1987. The Sweby region $\mathcal{D}_{1}$ is sufficient for a one-dimensional scheme with flux limiters to be TVD, the Spekreijse region $\mathcal{D}_{2}$, which has two free parameters $\alpha \in (-\infty,0]$, $M \in (0,\infty)$, is also sufficient for the scheme to be TVD.
• Figure 2.1: \ref{['fig:flux form stencil']} Shows variable placement for a flux form scheme, consistent with numerical method \ref{['EQ:SemiDiscrete Method']}, where the velocity field is located at cell faces/edges. \ref{['fig: advective form stencil']} shows variable placement for an advective form scheme, consistent with \ref{['eq:advective form']}, where the velocity is located at the cell centre. One can use Spekreijse's limiter region using the advective form stencil, but not when using the flux form stencil (\ref{['thm:necessary']}). When using a flux form scheme, one can instead use the limiter regions presented in this paper \ref{['region:Woodfield']}, \ref{['thm: divergence']}.
• Figure 3.1: It is necessary (but not proven sufficient), that limiters for incompressible flux form advection are contained in the above regions for positivity preservation. For $\theta=0$, $\theta = 1$, respectively where $M_{\psi}\in[0,\infty]$, $m_{\psi}\in[-\infty,0]$.
• Figure 3.2: New incompressible flow limiter regions, for $\theta=0,1$ respectively. Light gray regions are sufficient for a local maximum principle (bounded by solid lines), and dark grey indicates a desirable second order region (bounded by dashed lines). Light dotted line denotes a 3rd order region in the finite difference sense for uniform flow.
• Figure 4.1: $\mathcal{D}_{1}$ is the Sweby region, $\mathcal{D}_{2}$, is the Spekreijse region, which has two free parameters $\alpha \in [-\infty,0]$, $M \in (0,\infty)$. We can see that the limiters, van Albada (red) \ref{['limiter: Van Albada']}, Ospre (blue) \ref{['limiter: Ospre']}, and ENO2 (green) \ref{['limiter: Eno2']} are not contained in the Sweby region $\mathcal{D}_{1}$ (or even the new incompressible flow limiter region $\mathcal{D}_{4}$), but are in the Spekreijse region $\mathcal{D}_{2}$ with values for $[M,\alpha]$ given by $[1.5,-0.5]$, $[1/2(1+\sqrt{2}),1/2(1-\sqrt{2})]$, $[1,-1]$ respectively. Only the Ospre scheme and its Spekreijse Region are shown in \ref{['fig:spekOspre']}.

Theorems & Definitions (25)

• Definition 3.1: Spekreijse, Positive-coefficient type scheme S_1987
• Lemma 3.1: Jameson jameson1995
• proof
• Lemma 3.2: Spekreijse S_1987
• proof
• Remark
• Theorem 3.1: In-compressible flow
• proof
• Remark
• Theorem 3.2: Necessary conditions