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An optimization-based property-preserving limiter for two-phase flow in porous media on non-matching meshes
Sanghyun Lee, Seungmin Lee, Chen Liu
TL;DR
Saturation approximations in two-phase porous-media flow can violate physical bounds, and oscillations may persist near sharp fronts even when bounds hold. The paper combines mass-preserving cell-average optimization with targeted minmod limiting and Zhang-Shu scaling for DG discretizations on non-matching meshes. It proves fixed-time accuracy preservation for the cell-average limiter on those meshes.
Problem
Saturation approximations can violate physical bounds and develop spurious oscillations near sharp fronts, compromising physical consistency and numerical stability.
Source: §1 · pp. 1–2Method
A DG postprocessing limiter combines mass-preserving optimization of bounded cell averages, troubled-cell minmod limiting, and Zhang-Shu scaling of point values.
Source: §1 · p. 2Results
The cell-average limiter is proved to preserve numerical accuracy at a fixed time on non-matching meshes.
Source: §1 · p. 2Takeaways & Limitations
The framework extends bound and oscillation control for high-order DG approximations of two-phase flow to non-matching meshes.
Source: §1 · p. 2The model neglects gravity and assumes isotropic absolute permeability with bounded spatial variation.
Source: §2 · p. 3
Abstract
from arXiv · showhide
Two-phase flow in porous media arises in a wide range of applications in engineering and science. A fundamental challenge in its numerical simulation is that the approximation of the saturation transport equation can violate the maximum principle, and spurious oscillations can appear even when the solution remains within physical bounds. In this work, we propose a bound-preserving anti-oscillation postprocessing limiter for discontinuous Galerkin discretizations of the two-phase flow system on non-matching meshes. An optimization-based cell average limiter enforces the prescribed bounds on the cell averages while preserving global mass by minimizing the $L^2$ distance to the original cell averages. To suppress the remaining oscillations, the cell average limiter is combined with a minmod limiter, which is activated only in the cells identified by a troubled-cell detector to avoid unnecessary loss of accuracy, and a Zhang-Shu scaling limiter subsequently enforces the bounds on point values. In addition, we prove that the cell average limiter preserves the accuracy of the numerical solution at a fixed time on non-matching meshes. Numerical experiments on homogeneous, non-homogeneous, and fractured porous media demonstrate the accuracy and robustness of the proposed method.
1. Introduction
The paper addresses bound violations and spurious saturation oscillations in two-phase porous-media flow with a DG postprocessing limiter. It combines mass-preserving cell-average optimization, targeted oscillation control, and point-value scaling, extending the approach and its accuracy analysis to non-matching meshes.
1. Introduction
- Saturation approximations can violate physical bounds, producing unphysical solutions and numerical instabilities, especially in advection-dominated regimes with sharp fronts..Source: §1 · p. 1
- Existing bound-preserving strategies include convex-combination methods, matrix-monotonicity approaches, and flux limiting, while high-order methods on unstructured meshes may lack monotonicity.Some optimization-based approaches also require costly solves or problem-specific solver design.Source: §1 · pp. 1–2
- The two-phase model couples pressure and saturation equations, and the paper uses IMPES to solve pressure implicitly and saturation explicitly.Two-phase porous-media flow is also used in petroleum, environmental, biotechnology, and chemical engineering applications.Source: §1 · pp. 1–2
- The proposed DG postprocessing framework combines an optimization-based cell-average limiter, troubled-cell minmod limiting, and Zhang-Shu scaling to enforce bounds and suppress oscillations.The cell-average optimization minimizes distance to the original averages while preserving global mass; minmod acts only in detected troubled cells.Source: §1 · p. 2
- The cell-average limiter is extended to non-matching meshes, where its accuracy is proved to be preserved at a fixed time.The extension also targets conservation, bounds, and oscillation control across non-matching meshes.Source: §1 · p. 2
2. Governing equations
The governing model describes wetting and non-wetting phases in a porous medium through mass-balance equations, with wetting-phase pressure and saturation as primary variables. It specifies permeability and boundary conditions, while neglecting gravity.
2. Governing equations
- The model represents wetting and non-wetting phases in a porous medium and gives their densities, saturations, and Darcy velocities.The primary variables are wetting-phase pressure and saturation.Source: §2 · p. 3
- Absolute permeability is isotropic and bounded, relative permeability depends on wetting saturation, and gravity effects are neglected.The model assumes constant phase densities and that all pores are filled with fluid.Source: §2 · p. 3
- The domain boundary is partitioned into pressure Dirichlet and Neumann portions and saturation inflow and outflow portions, with initial conditions supplementing the system.The inflow and outflow sets are defined using the sign of wetting-phase velocity dotted with the outward normal.Source: §2 · p. 3
3. Numerical Methods
The numerical method uses DG discretization with an IMPES strategy and second-order time marching. Its spatial formulation uses piecewise bilinear discontinuous polynomials and applies the pressure and saturation DG formulations on a mesh of quadrilateral elements.
3.1. Temporal discretization
- The temporal scheme uses second-order BDF time marching and decouples the coupled nonlinear system with IMPES for sequential pressure and saturation solves.The pressure equation is treated implicitly and saturation explicitly.Source: §3.1 · p. 4, §3 · p. 4
3.2. Spatial discretization
- The spatial DG formulation uses piecewise bilinear discontinuous polynomials on a quadrilateral mesh and discretizes both pressure and saturation equations.The stated method is the incomplete interior penalty DG method.Source: §3.2 · pp. 4–5, §3 · p. 4
- The pressure equation is formulated implicitly in DG, with pressure boundary-edge sets and a penalty parameter specified for the spatial discretization.The saturation equation is formulated explicitly in DG using the computed wetting-phase pressure approximation and flux.Source: §3.2 · pp. 4–6
- For BDF2 startup, the initial saturation is interpolated pointwise and its values are replicated for the first time level.The saturation approximation is represented in the same spatial DG space.Source: §3.2 · p. 6
4. Postprocessing procedure on uniform meshes
On uniform meshes, the postprocessing sequence first bounds cell averages, then suppresses residual oscillations, and finally bounds point values. The minmod step is restricted to detected troubled cells to reduce unnecessary accuracy loss.
4. Postprocessing procedure on uniform meshes
- The procedure applies an optimization-based cell-average limiter, a minmod anti-oscillation limiter, and finally a Zhang-Shu scaling limiter for point values.The first stage enforces bounds on the zeroth-order moment; the last adjusts higher-order moments.Source: §4 · p. 7
4.1. Optimization-based bound-preserving limiter on uniform meshes
- The cell-average limiter minimizes distance to the original averages while preserving conservation and enforcing prescribed bounds.The minimizer is computed with generalized Douglas–Rachford splitting.Source: §4.1 · p. 7
- The fixed-time accuracy of the optimization-based cell-average limiter is established in prior work [41].This paper extends the algorithm to non-matching meshes.Source: §4.1 · p. 8
4.2. Minmod limiter on uniform meshes
- The minmod limiter targets residual oscillations, while a troubled-cell detector confines its use to cells with high-frequency components or sharp variations.Limiting only detected cells reduces unnecessary loss of accuracy; the chosen parameter is the most diffusive admissible option in the experiments.Source: §4.2 · pp. 8–9
5. Postprocessing procedure on non-matching meshes
The method extends the bound-preserving and minmod limiters to fixed non-matching meshes and establishes an accuracy result for the cell-average limiter. It adapts minmod neighbor information to coarse, fine, and equal-level interfaces.
5. Postprocessing procedure on non-matching meshes
- Both the optimization-based bound-preserving limiter and minmod limiter are reformulated for non-matching meshes, and an accuracy result is established for the cell-average limiter.The analysis focuses on a given non-matching mesh rather than dynamic mesh adaptivity.Source: §5 · p. 9
5.1. Optimization-based bound-preserving limiter on non-matching meshes
- The non-matching-mesh cell-average problem combines conservation and bound constraints; its convex feasible sets yield a strongly convex minimization problem with a unique solution.The cell-volume-weighted objective accommodates non-uniform cell volumes.Source: §5.1 · pp. 10–12
5.2. Minmod limiter on non-matching meshes
- For minmod limiting, equal-level neighbors use their coefficients unchanged, coarse neighbors use coefficients restricted to the shared subface, and finer-neighbor coefficients are averaged.The coarse-cell treatment preserves consistency across hanging-node interfaces.Source: §5.2 · pp. 12–13
6. Numerical experiments
The experiments test convergence and limiter behavior on uniform and non-matching meshes in homogeneous, non-homogeneous, and fractured domains. They report recovered optimal convergence rates and stable, bounded saturation solutions with oscillations controlled by the proposed limiting procedure.
6.1. Example 1: Convergence test
- Optimal convergence rates return from the second refinement cycle with BP and OF limiters on both uniform and non-matching meshes.On the coarsest uniform mesh, OF is triggered once and increases the errors; on the non-matching mesh, it initially produces a larger convergence rate.Source: §6.1 · p. 14, Table 2 · p. 14, Table 3 · p. 15
6.2.1. Example 2-1: Homogeneous domain
- In the homogeneous case, BP alone leaves sharp-front oscillations, whereas BP+OF suppresses them while matching the unlimited solution’s front locations.This pattern is reported for both uniform and non-matching meshes.Source: §6.2.1 · pp. 16–17
- On the homogeneous non-matching mesh, troubled cells shift with the sharp front and become fewer over time; unmarked regions retain sharper, less diffusive features.The detector identifies troubled cells in both coarse and refined regions early in the simulation, then limiting is needed locally near the remaining front.Source: §6.2.1 · pp. 17–18, Figure 10 · p. 18, Figure 11 · p. 18
6.2.2. Example 2-2: Non-homogeneous domain
- Without limiters, the non-homogeneous simulation develops oscillations and out-of-bounds values before crashing; BP+OF keeps saturation bounded and reaches the final time.On the non-matching mesh, BP+OF also suppresses oscillations while keeping saturation bounded.Source: §6.2.2 · p. 19, Figure 12 · p. 19, Figure 14 · p. 20
- In the non-homogeneous case, the lower detector threshold marks more cells and yields profiles without visible oscillations, while the higher threshold leaves oscillatory behavior.The same threshold comparison on a non-matching mesh reports more detected cells at the lower threshold and enhanced oscillation control.Source: §6.2.2 · p. 19, Figure 13 · p. 20, Figure 15 · p. 21
6.3. Example 3: Flow in fractured domain
- In the fractured domain, saturation advances first through high-permeability fractures and then into the matrix; detector-based limiting preserves sharper fronts near fracture interfaces with stable, oscillation-free profiles.The detector-based solution is reported as very similar to limiting everywhere, but less diffusive within fractures and near fracture-matrix interfaces.Source: §6.3 · p. 21, Figure 18 · p. 22, Figure 17 · p. 22
7. Conclusion
The proposed DG postprocessing framework combines bound and mass preservation with targeted oscillation control, and is reported to retain optimal convergence while performing robustly across several porous-media settings.
7. Conclusion
- Numerical experiments report optimal convergence rates, conservation and physical bounds, sharp saturation fronts, and robust performance across homogeneous, non-homogeneous, and fractured media.The reported results include both homogeneous and non-homogeneous porous media and fractured domains.Source: §7 · p. 23
- The optimization-based cell average limiter enforces prescribed bounds and global mass by minimizing the L2 distance to original averages, with DY splitting used to compute the minimizer efficiently.Source: §7 · p. 23
- A minmod limiter suppresses spurious oscillations, while a troubled-cell detector restricts limiting to troubled cells to avoid unnecessary numerical diffusion.Source: §7 · p. 23
- For non-matching meshes, the work establishes matrix-vector-level nonexpansiveness and identifies this framework as its first extension to such meshes.Source: §7 · p. 23