The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

Arthur Sarthou (1) , Stéphane Vincent (2) , Philippe Angot (3) , Jean-Paul Caltagirone (2)

1 IMFT - Institut de mécanique des fluides de Toulouse
2 I2M - Institut de Mécanique et d'Ingénierie
3 I2M - Institut de Mathématiques de Marseille

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance , the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

Mots clés: Fictitious Domain; Immersed Interface Method; Immersed Boundary Method; Penalty Methods; Finite Volumes; Elliptic Equations; Jump Embedded Conditions

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]; Modélisation et simulation

Arthur Sarthou, Stéphane Vincent, Philippe Angot, Jean-Paul Caltagirone. The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions. Open Journal of fluid Dynamics, 2020, 10 (3), pp.239 - 269. ⟨10.4236/ojfd.2020.103015⟩. ⟨hal-02921726⟩