Conservative Level-Set Method Review

• The conservative level-set method is defined by a bounded, smooth phase indicator that preserves interface geometry and enhances mass conservation in multiphase flows.
• It employs integrated advection and reinitialization techniques—such as redistancing and global mass correction—to maintain the signed-distance property and capture interface curvature.
• Key applications include incompressible flows, capillary dynamics, and immersed-boundary simulations, with implementations in finite-element, finite-volume, and hybrid frameworks.

Searching arXiv for recent and foundational papers on conservative level-set methods. The conservative level-set method is a class of interface-capturing formulations in which the moving interface is represented by a bounded diffuse variable—commonly a regularized Heaviside or hyperbolic-tangent profile—and evolved so that interface geometry remains tractable while phase conservation is improved relative to the standard level-set method. In representative formulations, the interface is the $\alpha=1/2$, $\psi=0.5$, or $\phi=0.5$ isocontour; advection is supplemented by reinitialization, embedded redistancing, or a posterior mass correction in order to preserve interface thickness, restore signed-distance structure, or enforce global mass balance (Waclawczyk, 2015, Papillon-Laroche et al., 30 Jan 2026). A major line of work also shows that, for incompressible flow, the advection and reinitialization steps can be combined into a discretely conservative and bounded phase-field equation, including variants with provable boundedness under specific central-difference discretizations (Mirjalili et al., 2018).

1. Interface representation and governing idea

A defining feature of conservative level-set formulations is the use of a bounded interfacial variable that behaves like a smooth phase indicator while remaining tied to a signed-distance-like description of the interface. In one common form, the conservative level-set function is

$$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$

where $\psi_0$ is the signed distance function and the interface is at $\alpha=1/2$ (Waclawczyk, 2015). Closely related formulations write

$$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$

or equivalently

$$\psi(\mathbf{x},t)=\frac{1}{1+\exp\!\left(-\frac{\phi(\mathbf{x},t)}{\varepsilon}\right)},$$

with $\phi$ the signed-distance field and $\varepsilon$ the interface-thickness parameter; in these cases the interface is the $\psi=0.5$0 contour (Boukharfane, 2022, Parameswaran et al., 2020).

This bounded representation distinguishes conservative level-set methods from classical signed-distance transport in two ways. First, it regularizes the interfacial jump over a narrow band, which makes property interpolation and curvature evaluation straightforward in one-fluid formulations. Second, it admits conservative transport laws for the smoothed phase indicator itself. In monolithic finite-element formulations, the transported quantity is often the smoothed sign or Heaviside transform $\psi=0.5$1, so that conservative phase transport and distance-function regularization are solved together (Luna et al., 2019).

The method is therefore best viewed not as a single algorithm but as a family of formulations sharing three structural ingredients: a diffuse but controlled interfacial profile, a transport step that respects phase conservation at the discrete level as far as possible, and a geometric restoration mechanism—reinitialization, redistancing, or global correction—that maintains the usefulness of the level-set field for normals, curvature, and coupling to the momentum equations (Papillon-Laroche et al., 30 Jan 2026).

2. Classical conservative reinitialization and its signed-distance interpretation

The classical conservative reinitialization mechanism balances a compressive flux against a diffusive flux in pseudo-time. One representative form is

$\psi=0.5$2

where $\psi=0.5$3, $\psi=0.5$4 and $\psi=0.5$5 are constants, and $\psi=0.5$6 is artificial time (Waclawczyk, 2015). Another equivalent CLS form evolves the phase field in pseudo-time as

$\psi=0.5$7

whose steady solution is the hyperbolic-tangent profile

$\psi=0.5$8

with $\psi=0.5$9 the signed distance to the interface (Papillon-Laroche et al., 30 Jan 2026).

A central theoretical development is the reinterpretation of the conservative variable as a localized signed-distance representation. The relation

$\phi=0.5$0

gives an exact signed-distance relation for the stationary CLS profile, and the reformulated reinitialization equation becomes

$\phi=0.5$1

which makes explicit that steady state is attained when $\phi=0.5$2 and the phase indicator remains localized near the interface (Waclawczyk, 2015). The same work shows that the reinitialization and advection equations of the conservative level-set function are mathematically equivalent to the corresponding equations for a localized signed distance function, and that a consistent derivative discretization yields second-order convergence of interface curvature on gradually refined grids (Waclawczyk, 2015).

This signed-distance interpretation matters because the conservative level-set field is not merely a smoothed volume fraction. Its utility depends on preserving a distance-like structure near the interface, especially when normals, curvature, and surface-tension forces are reconstructed from the same field. In that sense, reinitialization is not an auxiliary cosmetic step but a constitutive component of the method.

3. Boundedness, consistency, and reformulated reinitialization

A major concern in conservative level-set design is whether the bounded phase indicator remains within its admissible interval while avoiding excessive numerical dissipation. One important result is that, for incompressible flow, the advection and reinitialization steps of the conservative level-set scheme can be combined into a single phase-field equation similar to that of Chiu and Lin, and that for certain choices of the free parameters $\phi=0.5$3 and $\phi=0.5$4 a specific central-difference discretization guarantees boundedness of the phase field (Mirjalili et al., 2018). In that formulation, central finite differences are used in a setting where such schemes had long been avoided for two-phase advection because of overshoots and undershoots; the resulting equation is discretely conservative and bounded, free of any reinitialization or mass redistribution, and is presented as having competitive accuracy-vs-cost trade-off, small memory requirements, ease of implementation, and excellent parallelizability (Mirjalili et al., 2018).

A separate line of work addresses a different defect of standard artificial-compression reinitialization: curvature-dependent interface drift and patch formation away from the interface. The standard CLS reinitialization

$\phi=0.5$5

uses contour normals $\phi=0.5$6 from the initial reinitialization state. A reformulated alternative removes these normals entirely and replaces the pseudo-time dynamics by

$\phi=0.5$7

where the first term sharpens the interface and the second maintains the desired smooth transition region (Parameswaran et al., 2020). The exact CLS profile is a steady state of this new equation, the $\phi=0.5$8 contour remains fixed during reinitialization, and the absence of explicit contour normals suppresses the unphysical patch formation associated with ill-conditioned normals far from the interface (Parameswaran et al., 2020).

The reported numerical evidence reflects these design goals. In stationary reinitialization tests, no visible deformation occurs even after $\phi=0.5$9 pseudo-time iterations with the new formulation, whereas the classical Olsson-type scheme deforms interfaces progressively, especially where curvature is high (Parameswaran et al., 2020). The same study also reports a much larger allowable pseudo-time step,

$$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$0

and improved area conservation in rotating-disc, Zalesak-disk, dam-break, Rayleigh–Taylor, and rising-bubble tests (Parameswaran et al., 2020). Taken together with the consistency results of the signed-distance mapping approach, these developments show that conservative level-set research has focused not only on formal conservation but also on eliminating subtle geometry errors introduced by the reinitialization mechanism itself (Waclawczyk, 2015, Parameswaran et al., 2020).

4. Mass conservation, variational correction, and distance preservation

The classical motivation for conservative level-set methods is that the standard level-set method is inherently non-conservative. One analysis attributes this to two numerical sources: advection discretization error and reinitialization-induced interface shift (Long et al., 2022). A more recent variational analysis goes further and argues that the more fundamental culprit is the smooth Heaviside and delta regularization itself: even if reinitialization were exact and the zero contour remained stationary, the use of smooth interface distributions still violates mass conservation because the interfacial contribution does not cancel exactly in the diffuse layer (Khedkar et al., 2024).

This reinterpretation leads to constrained formulations. For two-phase flow, the mass constraint can be enforced by introducing a Lagrange multiplier into the advection equation, producing

$$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$1

with $$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$2 determined so that $$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$3 (Khedkar et al., 2024). As the regularized delta sharpens toward the Dirac delta, the exact Lagrange multiplier tends to zero, which indicates that the required correction is a diffuse-interface effect rather than a sharp-interface one (Khedkar et al., 2024). The drawback is that the exact correction acts nonuniformly across the interfacial band and destroys the signed-distance property restored by reinitialization.

That tension motivates uniform-shift strategies. A posterior mass correction can be written as

$$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$4

where the scalar $$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$5 is found from a Newton solve enforcing the prescribed phase volume (Long et al., 2022). Because $$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$6 is spatially uniform,

$$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$7

so the signed-distance property is preserved exactly after correction (Long et al., 2022). In the reported benchmarks, $$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$8–$$\alpha=\frac{1}{2}\left(1+\tanh\left(\frac{\psi_0}{2}\right)\right)=\frac{1}{1+\exp(-2N_c\psi_0)},$$9 Newton iterations are sufficient, the correction does not need to be applied at every time step, and using it every $\psi_0$0 steps is enough in the reported tests; the conservation error is reduced to the order of machine accuracy with negligibly extra cost (Long et al., 2022).

The same global-shift idea appears in the approximate Lagrange-multiplier method derived variationally. There, one first reinitializes to a signed-distance field $\psi_0$1 and then applies a uniform shift

$\psi_0$2

with $\psi_0$3 chosen so that

$\psi_0$4

This approximate multiplier is presented as a uniformized version of the exact variational correction and is extended to three-phase flow and fluid–structure interaction in both immersed and non-immersed forms (Khedkar et al., 2024). A plausible implication is that the modern conservative level-set literature distinguishes two different conservation problems: conservative transport of the diffuse phase field and exact preservation of the geometrically meaningful enclosed volume.

5. Discretization frameworks and coupled formulations

Conservative level-set methods have been realized in several numerical frameworks rather than a single canonical discretization. In unstructured finite elements, one monolithic formulation solves

$\psi_0$5

so that conservative transport of the smoothed indicator and signed-distance enforcement are handled in one PDE-like system (Luna et al., 2019). That method introduces a new PDE-based redistancing strategy for initialization from a simple phase indicator, an extra pre-redistancing stage at every time step to ensure convergence to the signed-distance level-set function, and a consistent algebraic artificial viscosity for the momentum equations with no tunable parameters; among the numerical parameters, only the dimensionless $\psi_0$6 is identified as strongly affecting performance, and the reported computations use $\psi_0$7 throughout (Luna et al., 2019).

In finite-volume settings, CLS is often coupled to a conservative transport variable. One example is a liquid–gas formulation implemented in MPflow, where the signed-distance field $\psi_0$8 is advected with a third-order WENO scheme and TVD third-order Runge–Kutta time stepping, periodically reinitialized by a local interface-aware procedure, and coupled algebraically to a VOF field $\psi_0$9 so that the accumulated mass conservation errors remain reasonably low (Lyras et al., 2021). The same paper emphasizes applicability to structured and unstructured meshes and extends the coupled method to Eulerian–Lagrangian spray atomisation through the ELSA model (Lyras et al., 2021).

A related hybrid route is the THINC/LS method, which uses the level-set field for local interface geometry and a THINC variable for conservative transport. The interface in each interfacial cell is represented by a polynomial surface of arbitrary order,

$\alpha=1/2$0

and a scalar correction $\alpha=1/2$1 is computed by Newton iteration so that the cell-averaged THINC function matches the conservative volume fraction exactly (Qian et al., 2018). In this framework, the level-set field is advanced by fifth-order Hamilton–Jacobi WENO and third-order TVD Runge–Kutta, while the THINC field is updated conservatively in finite-volume form (Qian et al., 2018).

Formulation family	Representative paper	Key mechanism
Monolithic CLS FEM	(Luna et al., 2019)	Conservative transport of $\alpha=1/2$2 plus signed-distance penalty
CLS/VOF coupling	(Lyras et al., 2021)	WENO-advection LS with algebraic VOF coupling
THINC/LS hybrid	(Qian et al., 2018)	Mass-constrained THINC volume fraction synchronized with polynomial LS geometry
Posterior conservative correction	(Long et al., 2022)	Global Newton shift $\alpha=1/2$3 preserving $\alpha=1/2$4

These implementations indicate that “conservative level-set method” is partly a representational notion and partly a coupling strategy. In some variants conservation is built into the transported field itself; in others it is restored by synchronization with a conservative auxiliary field or by a posterior global correction.

6. Applications, comparative results, and methodological boundaries

The method has been used in incompressible two-phase flow, capillary flows, atomisation, immersed-boundary formulations, and extrusion-based additive manufacturing. In extrusion-based 3D printing, for example, the conservative level-set equation is written as

$\alpha=1/2$5

and mass conservation is assessed through the strand cross-sectional area $\alpha=1/2$6 relative to the ideal area $\alpha=1/2$7 implied by the inlet volumetric flow rate (Rojas et al., 28 Aug 2025). That study reports that reducing the reinitialization parameter $\alpha=1/2$8 and the interface-thickness parameter $\alpha=1/2$9 generally decreases the cross-sectional error, but only up to a point; on a coarser mesh the error drops from about $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$0 to about $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$1 when $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$2 is tuned to $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$3, which suggests that an adequate interface thickness can reduce the strong mesh requirements often associated with level-set simulations (Rojas et al., 28 Aug 2025).

For capillary flows, a recent comparative study evaluates three reinitialization strategies inside a CLS solver: the original PDE-based CLS reinitialization, a new geometric reinitialization based on level-set redistanciation, and a projection-based sharpening method (Papillon-Laroche et al., 30 Jan 2026). The PDE-based and geometric methods both produce high-quality, spatially converged results for the rise of a bubble, capillary migration of a droplet, and the Rayleigh–Plateau instability, whereas the projection-based approach fails to capture complex $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$4D interfacial dynamics (Papillon-Laroche et al., 30 Jan 2026). The same comparison emphasizes a parameterization contrast: the geometric method uses only $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$5, while the PDE-based method requires the case-dependent selection of four parameters $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$6 (Papillon-Laroche et al., 30 Jan 2026).

Conservative level-set ideas have also been incorporated into immersed-boundary and ghost-fluid solvers. In an accurate conservative level-set strategy for incompressible flow around tandem cylinders, the solid–fluid boundary is represented by the $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$7 isocontour of a hyperbolic tangent profile, reinitialized with an inverse mapping

$$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$8

and supplied with FMM-based normals; the reported implementation performs one iteration of the improved reinitialization equation per time step with $$\psi(\mathbf{x},t)=\frac{1}{2}\left[\tanh\left(\frac{\phi(\mathbf{x},t)}{2\varepsilon}\right)+1\right]$$9 (Boukharfane, 2022). This suggests that CLS concepts remain useful even when the “interface” is a moving immersed boundary rather than a fluid–fluid free surface.

At the same time, the label has limits. A p-adaptive LDG level-set method for Willmore flow is described as energy stable and mass conservative, but it is explicitly not presented as a new conservative level-set formulation in the classic sense of multiphase interface capturing (Guo et al., 2016). The distinction is substantive: a method may conserve a discrete mass-like quantity while still lacking the conservative reinitialization, bounded phase-indicator transport, or mass-correction machinery associated with conservative level-set methods for incompressible multiphase flow.

Several recurrent controversies therefore persist. One concerns the source of mass loss: some analyses emphasize advection and reinitialization error, whereas the variational analysis identifies the smooth Heaviside/delta regularization as the main culprit (Long et al., 2022, Khedkar et al., 2024). Another concerns reinitialization itself: PDE-based schemes can preserve the desired $$\psi(\mathbf{x},t)=\frac{1}{1+\exp\!\left(-\frac{\phi(\mathbf{x},t)}{\varepsilon}\right)},$$0-profile but may induce interface displacement if parameters are poorly chosen, while exact variational corrections enforce mass but can destroy the signed-distance property (Papillon-Laroche et al., 30 Jan 2026, Khedkar et al., 2024). The contemporary literature increasingly treats these not as isolated defects but as coupled design constraints, with interface thickness, reinitialization strength, correction frequency, and geometric fidelity all entering the method’s practical definition (Mirjalili et al., 2018, Rojas et al., 28 Aug 2025).