Armando Coco

Armando Coco, Giovanni Russo

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations, leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway.

Armando Coco, Giovanni Russo

In this paper we present a multigrid approach to solve the Poisson equation in arbitrary domain (identified by a level set function) and mixed boundary conditions. The discretization is based on finite difference scheme and ghost-cell method. This multigrid strategy can be applied also to more general problems where a non-eliminated boundary condition approach is used. Arbitrary domain make the definition of the restriction operator for boundary conditions hard to find. A suitable restriction operator is provided in this work, together with a proper treatment of the boundary smoothing, in order to avoid degradation of the convergence factor of the multigrid due to boundary effects. Several numerical tests confirm the good convergence property of the new method.

Armando Coco, Alessandro Coclite, Stéphane Clain et al.

Unfitted boundary methods are widely used to numerically solve partial differential equations (PDEs) on irregular domains, avoiding the computational burden of generating boundary-conforming grids. In the finite-difference framework, structured Cartesian grids offer advantages such as ease of implementation and efficient parallelization, while geometry is represented implicitly, for instance, through level-set functions. In this setting, ghost point methods are commonly employed to enforce boundary conditions by introducing additional relations between interior and ghost nodes. However, constructing these relations becomes challenging for high-order accurate discretizations, which often rely on wide stencils that can reduce computational efficiency and degrade performance in large-scale parallel simulations. In this work, we investigate alternative ghost-point discretizations based on compact stencils. We introduce a formulation based on a boundary operator that locally approximates the boundary condition near each ghost node, replacing it with linear relations involving both interior and ghost points. The operator is constructed via least-squares reconstruction, allowing flexible stencil configurations while preserving the desired order of accuracy. Several strategies for selecting and adapting compact stencils are proposed, guided by conditioning criteria and iterative refinement procedures to improve global stability. Numerical experiments on various geometries and convection-diffusion regimes demonstrate the effectiveness of the proposed approach, showing that it maintains high accuracy even in the presence of boundary layers and improves stencil compactness and conditioning of the resulting linear systems.

Hridya Dilip, Clarissa Astuto, Armando Coco et al.

We develop a new numerical technique for approximating solutions of the Navier-Stokes equations on moving domains. The method aims at simulating an incompressible fluid past an object whose motion is assigned a priori using a level-set function. The proposed approach relies on a space discretization based on the ghost finite element method (ghost-FEM), which allows computations on unfitted meshes and avoids costly remeshing as the domain evolves in time. Time integration is performed using an IMplicit-EXplicit (IMEX) scheme to address the nonlinearity of the convective term, ensuring high-order accuracy for incompressible flows. The error introduced by the geometrical approximation is handled using the Shifted Boundary Method, which allows higher order approximations of boundary conditions on unfitted meshes. Dirichlet boundary conditions are imposed weakly by means of Nitsche's method. The associated stabilization parameter is chosen by solving a generalized eigenvalue problem, ensuring stability and accuracy of the numerical scheme. We present a series of numerical experiments designed to validate the accuracy of the proposed method, as well as comparisons with established benchmark problems involving moving boundaries.