Delta Redistribution: a variation of State Redistribution for steady-state embedded boundary computations

Background

• SRD stabilizes cut-cell evolution on embedded-boundary Cartesian grids and has been used in finite-volume and discontinuous Galerkin codes, including AMReX.Reported features include ease of implementation, high-order extensions, GKS stability for a 1D model problem, and a monotonicity-preserving variation [3].Source: §1 · p. 1

Motivation

• For steady-state computation, SRD changes the base scheme’s exact steady state by truncation error, so the base residual is nonzero at steady state.The paper says convergence must instead be assessed using max |q^n+1 − q^n|.Source: §1 · p. 1

Contribution

• DRD is a variation of SRD that converges exactly to the underlying scheme, unlike SRD’s truncation-error shift to steady solutions.The paper also reports GKS stability for a 1D linear-advection model, while noting DRD loses weighted SRD’s monotonicity property.Source: §1 · p. 1

Scope

• DRD is presented for cases where properties such as well-balancing should be preserved; the note previews smooth and shocked two-dimensional flow examples.The examples include a smooth-flow convergence comparison and steady and unsteady shocked flows.Source: §1 · p. 1

Merging formulas

• SRD forms convex combinations of provisional states, whereas DRD forms convex combinations of updates using the same weights.The merging formulas are the same in form but applied to the provisional state for SRD or the update for DRD.Source: §2 · p. 2

Merging neighborhoods

• A cut cell below the volume-fraction threshold temporarily combines with a neighboring cell in its most normal direction away from the embedded boundary.Neighborhoods may include multiple cells until they reach a volume threshold, typically half the full-cell volume.Source: §2 · p. 2

Provisional update

• The update D_i,j incorporates the entire update, while V_i,j is the cut-cell volume and A_k is the face length to a neighboring cell in the update stencil.The paper describes these quantities in its provisional cut-cell update.Source: §2 · p. 2

Overlap count

• The overlap count N_i,j counts how many merging neighborhoods contain a cell, with membership in one neighborhood reciprocally counted in the other cell’s overlap set.The formulas use the same convex-combination structure shown for SRD and DRD in Table 1.Source: §2 · p. 2, Table 1 · p. 2

Redistribution and variants

• After merging, the neighborhoods are distributed back to the Cartesian grid; DRD and SRD may use original or monotone weights, but DRD is not monotone with either set.The monotone weights reduce overshoot; the second-order version reconstructs a linear function through weighted centroids, and shocked-flow gradients are limited.Source: §2 · pp. 2–3, Table 2 · p. 3

3 Smooth Flows

• The smooth-flow tests use central-difference gradients on regular full cells and least-squares reconstruction on cut and irregular cells, without limiters.The reconstruction fits neighbors in a 3-by-3 fluid-cell neighborhood.Source: §3 · p. 3

3.1 Steady-state supersonic vortex

• For the steady vortex, linear and piecewise-constant DRD converge to steady state at the same rate, so linear redistribution is unnecessary.The paper reports this also saves over half the cost.Source: §3.1 · p. 3
• The supersonic-vortex density error converges at second order in the volume L1 norm and at order 1.5 on the boundary.The boundary norm weights density error at cut-cell centroids by embedded-boundary segment length.Source: §3.1 · pp. 3–4, Figure 1 · p. 4

3.2 Linear advection

• DRD-2 and SRD-2 have indistinguishable volume L1 errors, while DRD-1 has larger error but retains second-order volume convergence.The test integrates a cosine profile to time 2.0.Source: §3.2 · p. 4

4 Shocked Flows

• The examples comprise two steady shocked flows and one moving shock, with limiters specified for each case.The stated purpose is to demonstrate DRD’s performance with discontinuities.Source: §4 · p. 4

4.1 Regular Reflection

• For the steady regular reflection, DRD is used without gradients; limiter chatter begins after about 6 max-norm and 8 L1 orders of convergence.The case is a Mach 2.9 inflow on a 15° ramp.Source: §4.1 · p. 5, Figure 3 · p. 5

4.2 Steady Bow Shock

• At h=1/80, nose density and stagnation-pressure errors are 0.08% and 0.13%; Figure 4 shows smooth pressure contours at minimum volume fraction 0.14 × 10^-2.The half-cylinder case uses a Venkat limiter with K=1 on full cells and K=10 on cut cells.Source: Figure 4 · p. 6, §4.2 · p. 6

4.3 Unsteady Shocked Flow

• At t=2 on h=1/240, the Mach stem and contact are well-preserved; circular walls are represented by linear segments.The example is an unsteady Mach 10 shock on a 30° ramp.Source: §4.3 · p. 6, Figure 5 · p. 7

• DRD preserves steady-state solutions of the underlying base method and is cheaper than SRD because it needs no gradients.Source: §5 · p. 7
• With gradients, preliminary results show DRD is effective for time-dependent problems.Source: §5 · p. 7
• DRD lacks weighted SRD’s monotonicity bounds but is GKS stable for a linear advection model problem.The authors identify better understanding of the effects of limiting the update as future work.Source: §5 · p. 7

7 Appendix

• Compute weighted neighborhood centroids, then obtain merged values Q for DRD or SRD using Table 1.Source: §7 · p. 8
• Fit a linear function through merged values Q by least squares, using slopes fitted to the weighted neighborhood centroids.Source: §7 · p. 8
• For stability, the stencil should include at least one centroid no closer than h/2 in each direction; evaluate the reconstruction at the physical cell centroid.The evaluation step is linearity preserving.Source: §7 · p. 8