Level-Set Finite Difference Method

• Level-Set based Finite Difference Method is a numerical approach that implicitly defines interfaces using level-set functions, enabling dynamic tracking of complex geometries.
• It employs a PDE-based ghost point extension to accurately compute spatial derivatives near irregular boundaries, achieving third-order local accuracy.
• The method integrates with implicit time-stepping schemes, making it effective for simulating quantum systems, fluid dynamics, and other applications with evolving interfaces.

A level-set based finite difference method is a numerical framework in which interfaces and complex geometries are represented implicitly via level-set functions, and spatial derivatives required for finite difference approximations are computed directly on structured Cartesian grids. These methods are designed to combine the geometric flexibility of the level-set framework with the simplicity and efficiency of finite difference stencils, particularly for problems where the computational domain has curved or irregular boundaries or where interfaces undergo topological changes.

1. Level-Set Representation of Domains and Interfaces

In level-set based finite difference methods, the computational domain Ω is typically defined implicitly as the subset of a larger, regular computational grid for which a smooth level-set function φ(x) fulfills φ(x) < 0. The boundary ∂Ω corresponds to the zero level-set {x : φ(x) = 0}. The sign convention (negative inside, positive outside, or vice versa) is chosen for consistency with the application. The signed distance property (|∇φ| = 1 near the interface) is maintained for higher accuracy in normal and curvature computations.

This representation allows for seamless handling of moving interfaces and complex topologies without explicit interface tracking or mesh conformity. It also facilitates the extension of the method to multiple interacting interfaces and time-evolving domains.

2. Finite Difference Discretization and Ghost Point Extension

A central challenge in finite difference schemes on level-set defined domains is the treatment of grid points near the irregular boundary. Standard finite difference stencils cannot be applied directly at grid points adjacent to the interface, as some stencil points may lie outside Ω (so-called ghost points). To circumvent this, a PDE-based extension technique is employed:

• A constant extension (or Hamilton-Jacobi type equation) is solved in the vicinity of the interface:

$\partial_t u + \mathbf{n} \cdot \nabla u = 0$

where $\mathbf{n} = \nabla \phi / |\nabla \phi|$ is the normal vector at the interface.

• Ghost values are filled by evolving the above transport equation outward (from known values in Ω) using a first-order upwind or monotone scheme until convergence. This ensures that the function u is constant (or satisfies a specified symmetry property) along the normal direction away from the interface.
• For higher accuracy, a local Taylor series expansion is used:

$$u(\mathbf{x}_{g}) = \phi(\mathbf{x}_{g}) \partial_{n} u(\mathbf{x}^{*}_{g}) + \frac{1}{2} \phi(\mathbf{x}_{g})^2 \partial_{nn} u(\mathbf{x}^{*}_{g}) + O(h^3)$$

where $\mathbf{x}_{g}$ is a ghost point and $\mathbf{x}^{*}_{g}$ is its orthogonal projection onto the interface.

To further improve the extension, the technique is applied at the level of nodal basis functions defined on interior points, resulting in an explicit mapping from irregular point values to ghost point values. The resulting extension operator is incorporated into the finite difference matrix, allowing implicit time-stepping methods (such as backward Euler) to treat all unknowns, including those at ghost points, as part of the linear system without manual boundary treatments.

3. Normalized Gradient Flow for Ground State Computation

For applications such as computing the ground states of Bose-Einstein condensates, the method is integrated with a normalized gradient flow approach. The normalized gradient flow is implemented as a backward Euler finite difference time-stepping scheme:

$$\frac{\tilde{u}_{i,j}^{n+1} - u_{i,j}^{n}}{\Delta t} = \frac{1}{2} D_h \tilde{u}_{i,j}^{n+1} - V_{i,j} \tilde{u}_{i,j}^{n+1} - \beta |u_{i,j}^n|^2 \tilde{u}_{i,j}^{n+1}$$

followed by normalization

$$u^{n+1} = \tilde{u}^{n+1} / \|\tilde{u}^{n+1}\|_{2,h}$$

where $D_h$ is the finite difference Laplacian with proper treatment near the interface using ghost point extensions.

This framework naturally preserves the required $L^2$ mass constraint and allows for efficient and robust convergence to the ground state. It generalizes to other nonlinear problems, including higher-order and multi-component Bose–Einstein condensate models.

4. High-Order Accuracy Near Curved and Irregular Boundaries

The PDE-based extension technique is designed to provide third-order local accuracy for ghost point values near smooth boundaries, contingent on the smoothness and regularity of the interface (special care may be required for sharp corners or high-curvature regions). Central steps include:

• Construction of extension matrices $A_h$ (for basis function extension), $C_h$ (for curvature terms), $G_h$ (assembly of irregular points), and a diagonal matrix $\Phi_h$ (for normal distances).
• The final ghost value mapping is given by:

$$U_{\text{ghost}} = \Phi_h (A_h - \frac{1}{2} \Phi_h A_h C_h) G_h U_{\text{irregular}}$$

where $U_{\text{ghost}}$ and $U_{\text{irregular}}$ denote the vectors of ghost point and irregular point values.

This mapping is precomputed and re-used in the time-stepping, making the method computationally efficient for implicit schemes in complex domains.

5. Numerical Experiments and Performance

A suite of numerical examples demonstrates the effectiveness and versatility of the level-set based finite difference method:

• Laplace eigenvalue problem on non-convex (L-shaped) domains: The smallest eigenvalue converges at a rate matching theoretical predictions for non-smooth domains.
• Ground state computation in harmonic and box potentials in domains with curved boundaries (circles, ellipses, crescents): Accurate computation of ground state and excited state chemical potentials, matching analytical results where available.
• Problems with higher-order interaction terms (e.g., cubic-quintic nonlinearity, fractional Laplacians): The ghost extension operator is robust and accurate, even in challenging nonlinear regimes.
• Application to domains with corners or high curvature: Observed convergence rates degrade slightly near singularities, following theoretical expectations, but remain robust.

The method exhibits computational advantages due to the avoidance of body-fitted mesh generation and the efficient incorporation of the ghost value mapping into implicit time-steppers.

6. Advantages, Limitations, and Comparative Analysis

Advantages

• Geometric Flexibility: Naturally accommodates curved, non-convex, and even cornered domains without remeshing.
• High-Order Accuracy: Third-order accurate ghost point extension for smooth boundaries.
• Implicit Time-Stepping Compatibility: Ghost value extension is explicit in terms of interior unknowns, simplifying linear algebra for implicit and semi-implicit schemes.
• Automation: The extension operator for all required basis functions is computed once per geometry.
• No Interpolation Complexity: The technique eliminates the need for local stencil adaptation or manual interpolation procedures at boundary-adjacent points.

Limitations

• Accuracy at sharp corners or extremely high curvature can be limited; auxiliary cutoff or smoothing techniques may be needed.
• The method's extrapolation accuracy assumes smooth boundary geometry; in non-smooth settings, convergence rates are suboptimal by theory but still robust in practice.
• For highly nonlinear or stiff problems, further stabilization (e.g., convex-concave splitting) can be required.

7. Applications and Further Directions

Level-set based finite difference methods are broadly applicable in computational quantum systems (e.g., Bose–Einstein condensates), computational fluid dynamics (free boundary/interfacial flows), electromagnetics (wave propagation around curved objects), and beyond:

• In simulations involving multiphase boundaries or time-evolving domains, the method allows seamless extension without re-meshing procedures.
• The framework is extensible to higher spatial dimensions, more general boundary conditions, and coupling with adaptive mesh refinement strategies.
• For additional efficiency, the precomputed extension operator can be integrated with fast solvers or parallel processing architectures.
• Further improvement could target corners and high-curvature regions using curvature-aware stencils or hybrid cutoff approaches.

In summary, the level-set based finite difference approach delivers a robust, high-order, and efficient methodology for simulating PDEs on domains with complex, implicitly defined boundaries (Lee et al., 31 Aug 2025). Its automated ghost point extension and compatibility with implicit solvers make it highly suitable for problems where geometric flexibility and accuracy near irregular interfaces are critical.