Higher Order Multidimensional Slope Limiters with Local Maximum Principles

TL;DR

The paper tackles the challenge of enforcing local maximum principles for higher-order finite-volume schemes on unstructured meshes. It extends the Zhang slope-limiter framework to derive two local-boundedness limiters, $N(K)\cup K$-MP and $N^{2}(K)\cup N(K)$-MP, ensuring the next-time-cell mean $\bar{u}_K^{n+1}$ remains within locally defined bounds via a decomposition into flux-contributing quadrature points and monotone three-point Riemann solves. The authors apply these limiters to a second-order FV scheme (FV2) and develop a fourth-order FV method (FV4) that uses high-order projections, gradients, and Gauss quadrature, showing FV4 can retain 3rd–4th order accuracy in practice while maintaining a local maximum principle with the $N^{2}(K)\cup N(K)$-MP limiter. Numerical experiments on diverse 2D flows demonstrate the $N^{2}(K)\cup N(K)$-MP limiter generally outperforms or matches Barth–Jespersen in preserving local bounds with less severe limiting, whereas the $N(K)\cup K$-MP limiter can be more diffusive and reduce order for FV2. Overall, the framework provides a rigorous, scalable path to combining high-order accuracy with provable local boundedness on unstructured meshes, with implications for extensions to DG methods and steady-state computations.

Abstract

Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations and equations of transport type. Limiting is required to either converge to the correct type of solution or to adhere to physically motivated local maximum principles. Less restrictive limiting procedures are required so as to not severely decrease the accuracy. In this paper, we develop an existing slope limiter framework, to achieve different local boundedness principles for higher-order schemes on unstructured meshes. Quadrature points contributing to numerical fluxes can be limited based on face defined maximum principles, and the resulting cell mean at the next timestep can satisfy a cell mean maximum principle but with less limiting. We demonstrate the practical application of the introduced framework to a second-order finite volume scheme as well as a fourth-order finite volume scheme, in the context of the advection equation.

Figures (9)

• Figure 1.1: Diagram of cell $K$, and the face $\sigma_{KL}$ of a face sharing neighbour $L \in N(K)$, with outward unit normal $\boldsymbol n_{KL}$.
• Figure 2.1: Visualisations of some neighbourhoods, for some common meshes. Blue dot is an informal representation of the "middle" of the neighbourhood. Dark grey denotes the cells contained in the neighbourhood of the midpoint. Blue and orange regions are specific to the Barth and Jespersen limiter.
• Figure 3.1: The stencil $N(K)\cup N(L)$ for a structured and unstructured mesh. In particular this region is employed by the $N^2(K)\cup N(K)$-MP limiter for the second order finite volume scheme when ensuring that both $p_{K}(x_{KL})$ and $p_{L}(x_{KL})$ are locally bounded by surrounding cell means.
• Figure 3.2: Solid body rotation of the LeVeque initial conditions at $100\times 100$ resolution, using SSP22 timestepping with limiters at each internal substage of the Shu Osher representation. \ref{['fig:SSP22N1']} is the $N(K)\cup K$-MP limiter. \ref{['fig:SSP22BJ']} is the Barth and Jespersen limiter. \ref{['fig:SSP22N2']} is the $N^2(K)\cup N(K)$-MP limiter. \ref{['fig:SSP22KUZ']} is the Kuzmin/Park vertex limiter.
• Figure 3.3: Log-log plot of relative error of the SSP22 multidimensional limiters in $L^{2}$ for the smooth cosine bell initial conditions but different velocity fields. The velocity fields are defined by the stream functions \ref{['test: diagonally constant', 'test:quadradic reversing deformation', 'test:sin reversing deformation', 'test:solid body rotation constant']}.

Theorems & Definitions (16)

• Definition 1.1: Forward Euler Upwind
• Definition 1.2
• Example 1.1
• proof : Direct computation
• Theorem 1.1: Forward Euler HHLK monotone HHL_1976
• proof
• Remark
• Theorem 2.1: Monotone DG and FV schemes (with flux contributing vertex exclusion)
• Remark
• proof