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A new provably stable weighted state redistribution algorithm

Marsha Berger, Andrew Giuliani

TL;DR

The paper introduces a principled weighted state redistribution (SRD) framework for finite-volume updates on cut cells that is provably conservative, monotone in many common configurations, TVD, and GKS stable. By allowing the merge weights to continuously approach zero as the targeted cut-cell volume is approached, the method generalizes original SRD and prior continuous-cutoff variants, and clarifies when monotonicity can be guaranteed, particularly with pre-merging. The authors derive explicit weight formulas, prove conservation, and provide detailed monotonicity analyses for 2D linear advection (including normal and central merging) while extending the approach to 3D geometries via numerical experiments. Numerical tests in 2D and 3D Euler flows show reduced diffusion and robust stability across complex geometries (supersonic vortex, shock diffraction, trefoil cavity), with second-order convergence in smooth regions and maintained accuracy near embedded boundaries. The work highlights practical benefits—local, largely formula-based stabilization without global pre-processing—and outlines directions for extending monotonicity guarantees, reducing post-processing when updates vanish, and achieving higher-order accuracy.

Abstract

We propose a practical finite volume method on cut cells using state redistribution. Our algorithm is provably monotone, total variation diminishing, and GKS stable in many situations, and shuts off continuously as the cut cell size approaches a target value. Our analysis reveals why original state redistribution works so well: it results in a monotone scheme for most configurations, though at times subject to a slightly smaller CFL condition. Our analysis also explains why a pre-merging step is beneficial. We show computational experiments in two and three dimensions.

A new provably stable weighted state redistribution algorithm

TL;DR

The paper introduces a principled weighted state redistribution (SRD) framework for finite-volume updates on cut cells that is provably conservative, monotone in many common configurations, TVD, and GKS stable. By allowing the merge weights to continuously approach zero as the targeted cut-cell volume is approached, the method generalizes original SRD and prior continuous-cutoff variants, and clarifies when monotonicity can be guaranteed, particularly with pre-merging. The authors derive explicit weight formulas, prove conservation, and provide detailed monotonicity analyses for 2D linear advection (including normal and central merging) while extending the approach to 3D geometries via numerical experiments. Numerical tests in 2D and 3D Euler flows show reduced diffusion and robust stability across complex geometries (supersonic vortex, shock diffraction, trefoil cavity), with second-order convergence in smooth regions and maintained accuracy near embedded boundaries. The work highlights practical benefits—local, largely formula-based stabilization without global pre-processing—and outlines directions for extending monotonicity guarantees, reducing post-processing when updates vanish, and achieving higher-order accuracy.

Abstract

We propose a practical finite volume method on cut cells using state redistribution. Our algorithm is provably monotone, total variation diminishing, and GKS stable in many situations, and shuts off continuously as the cut cell size approaches a target value. Our analysis reveals why original state redistribution works so well: it results in a monotone scheme for most configurations, though at times subject to a slightly smaller CFL condition. Our analysis also explains why a pre-merging step is beneficial. We show computational experiments in two and three dimensions.
Paper Structure (17 sections, 71 equations, 11 figures, 1 table)

This paper contains 17 sections, 71 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Simple Model Problem with Weights
  3. The Weighted Algorithm and Conservation
  4. Monotonicity for Linear Advection
  5. First order linear advection in 2D
  6. Second order linear advection
  7. Numerical experiments
  8. Supersonic Vortex 2D Convergence Test
  9. Shock diffraction
  10. Acoustic pulse through trefoil
  11. Conclusions
  12. Monotonicity of Linear Advection
  13. Derivation of $\widehat{Q}$ Update Formula
  14. Monotonicity Proofs
  15. Proof of Property 3
  16. ...and 2 more sections

Figures (11)

  • Figure 1: Model problem with one small cell that merges to the right.
  • Figure 1: Merging normal to a planar embedded boundary can result in one of these two possible configurations. In both cases cell 0 has a volume fraction less than 1/2 and an overlap count of 1. Cell 1 has an overlap count of 2 and a volume fraction in larger than 1/2, the target merging volume, so it does not merge with any other cells.
  • Figure 1: Supersonic vortex test case. Left - density profile of test case. Zoom shows the size of the final updates ($N_i$). On this mesh only one cell has update size 3, with three cells contributing to its final update $U^{n+1}$. Right - convergence results for normal merging, comparing the new weights and the original weights, with both first order and second order accurate gradient reconstruction. Both the volume norm and boundary norm of the errors are presented.
  • Figure 1: Notation for the two possible configurations in 2D with SRD using normal merging for cell $C_0$ and flow parallel to a planar boundary.
  • Figure 1: Maximum root of GKS determinant is $\leq 1$ for the model problem with one small cell, merging left and without pre-merging. Left figure shows the original weights ($w=1/2$), right shows the new weights ($w=\alpha$). All growth is bounded by 1, showing stability of SRD although neither version is monotone. \newlabelfig:gksfig0
  • ...and 6 more figures