Physical and numerical models


Fig. 1: general configuration
Figure 1: general configuration

Configurations  supported by PROSPERO consit of  physical media Ωi separated by interfaces Γ as represented on figure 1. High order numerical methods are developped to simulate the propagation of mechanical waves in these heterogeneous media.

Physical Models

Numerous constitutives laws and dissipation terms are available for the media Ω :

  • Acoustics :  fluids 
  • Elasticity : isotropic or anisotropic solids
  • Viscoelasticity : Zener models
  • Poroelasticity :  Biot and JKD models including fractional dissipative mechanisms
  • Nonlinear elasticity : Vakhenko's models, viscoelasticity, slow dynamics 

Different conditions can be defined at the interfaces Γ :

  • Free surface
  • Perfect contact
  • Imperfect contact : bonding, hydraulic,...


PROSPERO is based on time-domain methods, where the equations are written as a general first-order velocity-stress formulation 

                       ∂t U + ∑i ∂x Fi(U) = S(U)

  • U is the vecteur of unknowns (velocity and stress components)
  • Except for the nonlinear laws, the physical fluxes Fi(U) writes as matrix-vector products. Matrices coefficients depend on the physical parameters of the media.
  • The general source term S(U) represents the attenuation of the wave. Its form depends on the choosen dissipation mechanisms (Zener, fractional, ...)

Détails can be found in the papers listed in the  publications  section. 


Numerical methods

The computational domain is  approximated with a Cartesian grid where High Order Finite Differences  schemes are  easily 

implemented and very efficients.

In counterparts an accurate representation of the interfaces is tricky but reached using the ESIM immersed interface method.

Discretization scheme

Discretization of the general partial differential system is based on Strang Splitting to separate the treatment of the propagative (homogeneous) part and  of the dissipative part (source term). The solution of the propagative part is approximated using a forth order (in space and time) ADER scheme . The differential system coming from the dissipative part is solved exactly.  

Special attention has been paid to develop an original and efficient approximation of the time fractional derivatives in the porous BIOT-JKD model.

Interface method

The interfaces separating the different media are treated with the Explicit Simplified Immersed Interface Method (ESIM) developped initially by B. Lombard and adapted to the different constitutives laws and contacts. The method is based on r-th order polynomial extrapolations of the solution through the interface.

Main ingredients and properties :

  • incorporation of the geometrie of the interface (relative position in the grid, curvature)
  • incorporation of the physical jump conditions corresponding to the contact
  • incorporation of the conservation laws 
  • conservation of the 4th order of the discretization scheme
  • negligible CPU extra cost (pre-processing step)


The resulting algorithm is explicit and the stability condition is optimal with CFL condition equals to 1.

Implementation has been carefully optimized in regards to the constitutives laws. 

MPI parallelization allows large speed-up for complex 3D configurations.