Sharp-Interface Immersed Boundary Method

• Sharp-interface immersed boundary method is a numerically precise technique that treats interfaces as distinct boundaries, enforcing jump conditions explicitly.
• It employs mechanisms like finite-difference corrections, ghost-cell reconstructions, and discrete forcing to maintain sharp pressure and stress discontinuities.
• The method facilitates accurate simulation of complex geometries and fluid–structure interactions while achieving second-order or higher convergence.

Sharp-interface immersed boundary method denotes a class of immersed discretizations in which a boundary or interface is represented as a geometrically distinct surface embedded in a non-body-fitted background grid, and the associated wall constraints or interfacial jump conditions are enforced sharply rather than regularized over several cells. In this literature, sharpness is realized through several related mechanisms: immersed interface corrections to finite-difference stencils, ghost-cell and image-point reconstruction, discrete forcing localized to stencil entries that cross the boundary, and hybrid Lagrangian–Eulerian constructions that retain an explicit interface while solving the field equations on fixed Eulerian meshes. The defining contrast with conventional regularized immersed boundary formulations is that pressure, viscous stress, traction, species, or thermal discontinuities are treated as actual jumps or boundary data, not as smeared source layers (Kolahdouz et al., 2018, Luchini et al., 17 Jun 2025, Ji et al., 20 Aug 2025).

1. Concept and defining characteristics

In conventional immersed boundary formulations, interfacial forces are spread to the Eulerian grid with regularized delta kernels and velocities are interpolated back with the same smoothing operators. Several papers identify the central limitation of that strategy: pressure and viscous stress are generally discontinuous at the interface, so regularization smooths the very quantities that determine interfacial loading and wall accuracy. Sharp-interface methods retain the geometric flexibility of non-body-fitted grids, but replace distributed smoothing by explicit interface treatment, typically at irregular grid points or cut stencils (Kolahdouz et al., 2018, Puelz et al., 2019).

The term covers more than one numerical lineage. In immersed interface methods, the singular force supported on a codimension-one interface is converted into jump conditions, and finite-difference operators near the interface are corrected accordingly. In ghost-cell methods, cells inside the body but adjacent to fluid cells provide auxiliary unknowns or reconstructed values so that the actual boundary condition is enforced at a body-intercept point. In discrete-forcing variants, the effect of the true wall position is absorbed into the local discrete operator, so that the wall remains sharp without solving on a body-fitted mesh. In vorticity–velocity formulations, the same sharpness is obtained by constructing extended values across the interface instead of adding diffuse body forces (Ji et al., 2023, Luchini et al., 17 Jun 2025).

Sharp-interface immersed methods are therefore best understood as a numerical principle rather than a single algorithm: the interface is treated as an actual boundary or jump set, while the ambient discretization remains Cartesian or otherwise background-grid-based. This permits complex static, moving, or deforming geometry without remeshing, but it also transfers substantial responsibility to the local reconstruction, extrapolation, or jump-projection machinery.

2. Governing equations and interface enforcement

A representative incompressible formulation writes the fluid equations as

$$\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{u}, \qquad \nabla \cdot \mathbf{u} = 0,$$

with an immersed interface $\Gamma_t$ across which the velocity is continuous,

$\llbracket \mathbf{u} \rrbracket = 0,$

while pressure and velocity gradients satisfy jump conditions induced by the concentrated interfacial force $\mathbf{F}$,

$\llbracket p \rrbracket = - \jmath^{-1} \mathbf{F}\cdot \mathbf{n}, \qquad \mu \left\llbracket \frac{\partial \mathbf{u}}{\partial x_i} \right\rrbracket = (\mathbb{I}-\mathbf{n}\mathbf{n}^T)\,\jmath^{-1}\mathbf{F}\,n_i.$

In the immersed interface setting, these relations supply correction terms for finite-difference stencils that intersect the interface, so that singular traction is imposed sharply and the Eulerian discretization can remain second-order accurate (Facci et al., 2024).

Ghost-cell formulations enforce the same idea through geometric reconstruction. A wall projection point $W$ and mirrored or image point $M$ are constructed from a ghost-cell center $G$, typically with

$\overrightarrow{GW}=\overrightarrow{WM}.$

The wall condition is enforced at the body-intercept point, while the image-point value is reconstructed from surrounding fluid states. For scalar Dirichlet data, one sharp reflection formula used in evaporation calculations is

$Y_{GC}=2Y_{BI}-Y_{IP},$

where $\Gamma_t$0 is the boundary-intercept point and $\Gamma_t$1 the image point on the fluid side. In high-order immersed-interface formulations, the corresponding boundary values are obtained from local least-squares polynomials and interface conditions, including Neumann data and material-jump relations (Salimnezhad et al., 2024, Ji et al., 20 Aug 2025).

A different sharp mechanism appears in discrete-forcing IBM. There the boundary is introduced by replacing the value at a ghost or external point with an extrapolated value consistent with the true wall location, and substituting that value directly into the discrete operator. In one such formulation, only the center coefficient of the Laplacian stencil in the momentum equation is modified, while no corrections are introduced for the continuity equation or the pressure equation (Luchini et al., 17 Jun 2025). The underlying commonality is that the physical boundary condition is imposed at the true interface location, not averaged over a forcing band.

3. Geometry representation, sharp features, and discontinuous jump data

A recurrent theme in sharp-interface research is that geometry representation is not secondary. An immersed interface method for discrete surfaces demonstrated that only a $\Gamma_t$2-continuous Lagrangian finite element surface is needed: the interface can be represented by standard nodal finite elements,

$\Gamma_t$3

and geometric or traction-related quantities can be projected onto the surface finite element space by an $\Gamma_t$4 projection. This eliminates the need for analytic surface parametrizations, analytic curvature, or globally smooth coordinates, while still allowing the method to sharpen pressure and stress jumps (Kolahdouz et al., 2018).

The difficulty is that a $\Gamma_t$5 geometry is not generally $\Gamma_t$6. For smooth bodies this is manageable, because element normals converge to the smooth normal under refinement. For corners, edges, and vertices, however, the normal is discontinuous, so the projected quantity $\Gamma_t$7 is also discontinuous and the jump conditions inherit that discontinuity. A 2024 extension of the immersed interface method showed that projecting such discontinuous jump data into a continuous Galerkin basis creates an $\Gamma_t$8 error at sharp corners. The paper therefore replaces the continuous Galerkin projection with a discontinuous Galerkin representation, in which the jump field is expanded in element-local basis functions without interelement continuity. For smooth interfaces, DG and CG have comparable accuracy; for sharp interfaces, DG avoids the artificial smoothing imposed by CG and yields stable time-step limits that are essentially insensitive to geometric sharpness (Facci et al., 2024).

A closely related issue arises in sharp-edged geometry for ghost-cell reconstruction. In all-speed viscous-flow simulations on triangulated surfaces, it was observed that classification and interpolation order alone are insufficient near sharp edges and vertices; the direction of reconstruction also matters. The remedy is to reconstruct along a local normal determined by the nearest geometric feature: face normal for a face, angle-weighted edge pseudo-normal for an edge, and angle-weighted vertex pseudo-normal for a vertex. This local-normal strategy, combined with ray tracing and robust closest-point queries, was shown to improve pressure fields, wake structure, and geometric fidelity near sharp trailing edges and wedges (Seshadri et al., 2019).

These results directly counter a common misconception that “sharp-interface” automatically implies faithful treatment of nonsmooth geometry. The literature instead shows that corners and edges require dedicated representation of discontinuous normals, discontinuous jumps, or pseudo-normal directions.

4. Numerical families and reported accuracy

The discrete-surface immersed interface method reports that only the lowest-order jump conditions for pressure and velocity gradient are required to realize global second-order accuracy. Its verification studies show second-order global convergence and nearly second-order local convergence for Eulerian velocity, between first- and second-order global convergence with first-order local convergence for Eulerian pressure, second-order local convergence in interfacial displacement and velocity, and first-order local convergence in fluid traction (Kolahdouz et al., 2018).

A different sharp strategy for codimension-0 immersed solids is pressure splitting. In a sharp-interface IBFE formulation, the physical pressure is decomposed as

$\Gamma_t$9

so that $\llbracket \mathbf{u} \rrbracket = 0,$0 is continuous on the whole computational domain and $\llbracket \mathbf{u} \rrbracket = 0,$1 is solved only on the solid. This removes the pressure jump from the Eulerian pressure solve and places it into a separate harmonic or diffusion problem on the solid. Reported effects include pointwise pressure convergence and substantially smaller errors for pressure, velocity, displacement, and stress than in conventional IBFE (Puelz et al., 2019).

At the opposite end of the algorithmic spectrum, a 2025 immersed-boundary method achieves second-order accuracy with a very compact correction: the solid-fluid interface is represented sharply by discrete forcing, the correction is implicit, and only the center weight of the Laplacian stencil in the momentum equation is modified. The pressure and continuity equations are left uncorrected. The method is verified on laminar and turbulent examples, including a sinusoidal channel wall and a human nasal cavity, with grid-convergence studies that are essentially second order (Luchini et al., 17 Jun 2025).

Sharp-interface ideas have also been extended to alternative formulations and higher orders. For the 2D vorticity–velocity Navier–Stokes equations, a sharp immersed interface method reaches second-order accuracy for most practical scenarios, and in a pitching-plate comparison it requires about $\llbracket \mathbf{u} \rrbracket = 0,$2–$\llbracket \mathbf{u} \rrbracket = 0,$3 times fewer grid points per dimension than a representative first-order penalization approach (Ji et al., 2023). At higher order, a fourth-order immersed interface solver combines a Runge–Kutta projection method with a fifth-order immersed advection discretization and reports fourth-order convergence of velocity and pressure in the infinity norm, both in the domain and on immersed boundaries, for stationary and one-way coupled moving geometries (Ji et al., 20 Aug 2025).

Taken together, these papers show that “sharp-interface immersed boundary method” does not imply a fixed order of accuracy. Second-order schemes remain common because of robustness and simplicity, but the framework now includes fourth-order projection methods, high-order immersed advection, and specialized pressure-splitting constructions.

5. Moving boundaries, fluid–structure interaction, and multiphysics

Moving-body problems introduce difficulties absent in stationary immersed geometries: grid points change type, “fresh” cells appear, and pressure or force signals can become contaminated by temporal reconstruction noise. One sharp-interface treatment for oscillating airfoils uses a ghost-cell-based field extension strategy, in which the flow field is extrapolated beneath the immersed surface so that nodes that later become fluid or immersed-boundary nodes already contain a smoothly extended solution. In the reported oscillating-cylinder test, the solution without field extension becomes chaotic and unphysical, whereas the extended-field version suppresses high-frequency oscillations in pressure and drag and matches benchmark data (Seshadri et al., 2021).

A related development is a fast and consistent sharp-interface IBM for moving bodies of arbitrary thickness. It combines a fast tagging algorithm, a two-sided Eulerian forcing strategy, and a consistent mass correction for fractional-step schemes, while preserving the discrete Laplacian operator and therefore the use of direct Poisson solvers. The method reports second-order accuracy in no-slip enforcement, small transpiration errors, automatic handling of moving and deformable bodies, and applicability to rigid, deformable, turbulent, and biologically inspired flows (Vagnoli et al., 8 Jun 2026).

For fluid–structure interaction, sharp-interface methods span both partitioned and hybrid immersed formulations. An implicit partitioned solver for an elastic splitter plate couples a sharp-interface immersed-boundary flow solver with a finite-element structural solver and uses Aitken-based dynamic under-relaxation; the revised coupling is reported to be around two to three times faster and numerically stable relative to constant under-relaxation (Kundu et al., 2020). A more general flexible-body immersed Lagrangian–Eulerian method solves distinct fluid and solid equations on their own subdomains, couples them through Dirichlet–Neumann interface conditions, and uses a penalty force

$\llbracket \mathbf{u} \rrbracket = 0,$4

as an approximate Lagrange multiplier. Its fluid solver relies on an immersed interface method for discrete surfaces and supports multi-rate time stepping (Kolahdouz et al., 2022).

Multiphysics variants extend sharp-interface methodology beyond no-slip walls. A hybrid immersed-boundary/front-tracking method for droplet evaporation uses ghost cells and image points to impose the vapor mass-fraction boundary condition sharply, reports overall second-order spatial accuracy, and extends the same machinery to mass transfer from a solid sphere (Salimnezhad et al., 2024). An immersed boundary–lattice Boltzmann wetting model represents the droplet as a sharp moving triangular mesh and introduces a wall-interaction force

$\llbracket \mathbf{u} \rrbracket = 0,$5

with an analytical relation between $\llbracket \mathbf{u} \rrbracket = 0,$6 and the equilibrium contact angle. The interface remains sharp globally while being regularized near the contact line (Bellantoni et al., 26 Mar 2025).

6. Regimes, performance, and unresolved technical issues

Sharp-interface immersed methods are now used across a wide range of regimes. Aerospace-oriented work includes a data-constrained sharp IBM for segregated OpenFOAM solvers with implicit time discretization, where a Luenberger observer updates localized source terms for velocity, density, and temperature; the reported strength is good pressure prediction and solver compatibility, while wall shear stress and viscous drag remain less accurate because the forcing focuses on field values rather than direct gradient constraints (Chemak et al., 23 Feb 2025). For high-speed compressible flows, a sharp-interface IBM integrated into blastFOAM enforces a slip-wall condition through quadratic reconstruction on an STL-defined immersed surface and is validated on wedges, cylinders, aerofoils, spheres, and a moving piston, with sharp shock resolution and good agreement with analytical and body-fitted results (Pandey et al., 17 Feb 2026).

Large-scale computing has become an explicit research direction. The GPU implementation of the sharp-interface solver ViCar3D ports ghost-cell classification, reconstruction, and halo exchange to multi-GPU architectures with OpenACC, CUDA, batched cuBLAS, and CUDA-aware MPI. For DNS past a finite rectangular wing it reports an approximately $\llbracket \mathbf{u} \rrbracket = 0,$7 node-to-node speedup relative to the CPU implementation, capability up to $\llbracket \mathbf{u} \rrbracket = 0,$8 million mesh points on a single four-GPU node, and maximum strong and weak scaling efficiencies of $\llbracket \mathbf{u} \rrbracket = 0,$9 and $\mathbf{F}$0 (Kumar et al., 22 May 2025). In spline-based immersed analysis, weighted quadrature, discontinuous weighted quadrature, and row-based assembly have been adapted to sharp cut-domain integration; the main bottleneck remains cut-element integration, although its fractional contribution declines with $\mathbf{F}$1-refinement (Marussig et al., 2023).

Several open technical issues recur across the literature. First, sharp interface treatment does not by itself resolve every near-wall quantity: some methods report that pressure loads are captured more accurately than viscous drag or wall shear. Second, moving interfaces remain algorithmically delicate, especially when fresh cells, stage-wise extrapolation, or mass correction interact with projection methods. Third, explicit front tracking preserves geometric sharpness but is less naturally suited to topological changes such as breakup and coalescence unless additional algorithms are introduced (Salimnezhad et al., 2024). Fourth, sharp geometric singularities remain a persistent source of $\mathbf{F}$2 projection error, pseudo-normal ambiguity, or time-step sensitivity unless the interface representation explicitly respects corners and edges (Facci et al., 2024, Seshadri et al., 2019).

The current trajectory of the field suggests an increasingly heterogeneous methodology rather than convergence to a single canonical algorithm. High-order immersed interface schemes, pressure-splitting formulations, direct-forcing variants that preserve simple linear algebra, sharp two-sided Eulerian methods for moving thin bodies, explicit interface-resolved multiphysics, and GPU-native solvers all coexist. A plausible implication is that “sharp-interface immersed boundary method” is now best regarded as a design space organized around one numerical commitment: non-body-fitted geometry should not require diffuse enforcement of boundary or interface physics.