Update-Magnitude State Redistribution (UM-SRD): A Shut-off Extension of Weighted SRD for Cut-Cell Methods

TL;DR

This paper introduces UM-SRD, an extension of weighted SRD for cut-cell methods that adaptively reduces dissipation near steady states, ensuring stability, accuracy, and exact steady-state preservation.

Contribution

UM-SRD blends SRD with the identity operator based on update magnitude, improving steady-state preservation and stability in cut-cell finite-volume schemes.

Findings

UM-SRD is total variation diminishing under the same CFL condition as the base scheme.

Numerical experiments show UM-SRD converges to first order on smooth 1D and 2D advection tests.

UM-SRD stabilizes the base scheme near small cut cells where divergence occurs.

Abstract

Berger & Giuliani (2024) developed a provably stable weighted state redistribution (SRD) algorithm for cut-cell meshes. A key limitation of their method is that, although flux redistribu- tion naturally vanishes when updates are small, SRD continuously applies redistribution even when the flux balance is zero, preventing exact steady-state preservation and potentially in- troducing unnecessary dissipation in smooth regions. This work introduces Update-Magnitude State Redistribution (UM-SRD), which blends the SRD operator with the identity operator via a smooth, locally-defined scalar indicator of the finite-volume update magnitude. UM-SRD preserves conservation and reduces exactly to the base scheme when the finite-volume update is exactly zero in a small-cell neighborhood. For a one-dimensional model problem with a single small cut cell, we prove UM-SRD is total variation diminishing under the same CFL condition as the base upwind scheme, show the local truncation error modification is higher-order in smooth regions with the unnormalized indicator, and show that the normalized implementation pre- serves first-order accuracy. Numerical experiments demonstrate convergence toward first order on smooth 1D and 2D advection tests, confirm shut-off behaviour, verify non-oscillatory proper- ties, provide numerical evidence that UM-SRD stabilizes the base scheme near a small cut cell where the base scheme diverges, and confirm exact steady-state preservation. The algorithm reuses existing weighted SRD infrastructure, adding only a local blending mechanism, making it practical for cut-cell finite-volume codes.