A second order finite-difference ghost-point method for elasticity problems on unbounded domains with applications to volcanology

Abstract

We propose a novel nite-di erence approach for the numerical solution of linear elasticity problems in arbitrary unbounded domains. The method is an extension of a recently proposed ghost-point method for the Poisson equation on bounded domains with arbitrary boundary conditions (Coco, Russo, JCP, 2013) to the case of the Cauchy-Navier equations on unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are de ned by suitable level-set functions. The equations are discretized by classical ninepoint stencil on interior points, while boundary conditions and high order reconstructions are used to de ne the eld variable at ghost-point, which are grid nodes external to the domain with a neighbor inside the domain. The approach is then adopted to solve elasticity problems applied to volcanology for computing the displacement caused by an underground pressure source. The method is suitable to treat problems in which the geometry of the source often changes (explore the e ects of di erent scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modi ed. Several numerical tests are performed, which asses the e ectiveness of the present approach. Keywords: Linear Elasticity, Cauchy-Navier equations, ground deformation, unbounded domain, coordinate transformation method, Cartesian grid, Ghost points, Level-set methods