We present a numerical method to take into account 2D arbitrary-shaped interfaces in classical finite-difference schemes, on a uniform Cartesian grid. This work extends the "Explicit Simplified Interface Method" (ESIM), previously proposed in 1D (2001, J. Comput. Phys. 168, pp. 227-248). The physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics, with stationary interfaces. Our method maintains, near the interfaces, properties of the schemes in homogeneous medium, such as the order of accuracy and the stability limit. Moreover, it enforces the numerical solution to satisfy the exact interface conditions. Lastly, it provides subcell geometrical features of the interface inside the meshing. The ESIM can be coupled automatically with a wide class of numerical schemes (Lax-Wendroff, flux-limiter schemes,...) for a negligible additional computational cost. Throughout the paper, we focus on the challenging case of an interface between a fluid and an elastic solid. In numerical experiments, we provide comparisons between numerical solutions and original analytic solutions, showing the efficiency of the method.

To cite this version:Joël Piraux, Bruno Lombard. A new interface method for hyperbolic problems with discontinuous coefficients: one-dimensional acoustic example. Equations of acoustics are written as a first-order linear hyperbolic system. Away from interfaces, a standard scheme (Lax-Wendroff, TVD, WENO...) is used in a classical way. Near interfaces, the same scheme is used, but it is applied on a set of modified values deduced from numerical values and from jump conditions at interfaces. It amounts to modify the scheme so that its order of accuracy is maintained at irregular points, despite the non-smoothness of the solution. This easy to implement interface method requires few additional computational resources and it can be applied to other partial differential equations.

accepted and to be published in Geophys. J. Int.International audienceA method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing

Abstract. The spring-mass conditions are an efficient way to model imperfect contacts between elastic media. These conditions link together the limit values of the elastic stress and of the elastic displacement on both sides of interfaces. To insert these spring-mass conditions in classical finitedifference schemes, we use an interface method, the Explicit Simplified Interface Method (ESIM). This insertion is automatic for a wide class of schemes. The interfaces do not need to coincide with the uniform cartesian grid. The local truncation error analysis and numerical experiments show that the ESIM maintains, with interfaces, properties of the schemes in homogeneous medium.

This paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids. Dissipation mechanisms are described using the generalized Zener model. No time convolutions are required thanks to the introduction of memory variables that satisfy local-in-time differential equations. By appropriately choosing the relaxation parameters, it is possible to accurately describe a large range of materials, such as solids with constant quality factors. The evolution equations satisfied by the velocity, the stress, and the memory variables are written in the form of a first-order system of PDEs with a source term. This system is solved by splitting it into two parts: the propagative part is discretized explicitly, using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is then solved exactly. Jump conditions along the interfaces are discretized by applying an immersed interface method. Numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems, such as multiple scattering configurations