A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

Coco, Armando; Currenti, Gilda; Negro, Ciro Del; Russo, Giovanni

doi:10.4208/cicp.210713.010414a

Cited by 26 publications

(19 citation statements)

References 37 publications

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“…In the case of fixed boundaries, our method is similar to the scheme from [2]. We, however, derive suitable boundary conditions for all field variables and use a different extrapolation technique, which is analogue to the one adopted in the context of elliptic [10] and elastic [9] problems. In particular, a 2-D interpolation which includes ghost points is used in order to impose boundary conditions on the projection of the ghost point values onto the boundary.…”

Section: Introductionmentioning

confidence: 99%

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

Chertock¹,

Coco²,

Kurganov³

et al. 2018

CICP

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In this paper, we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domain Ω, possibly with moving boundary. The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls. Although the motion of the boundary could be coupled with the fluid, all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow. The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domain Ω R ⊃ Ω. We identify inner and ghost points. The inner points are the grid points located inside Ω, while the ghost points are the grid points that are outside Ω but have at least one neighbor inside Ω. The evolution equations for inner points data are obtained from the discretization of the governing equation, while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables (density, velocities and pressure). Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains. Several numerical experiments are conducted to illustrate the validity of the method. We demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.

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Section: Introductionmentioning

confidence: 99%

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

Chertock¹,

Coco²,

Kurganov³

et al. 2018

CICP

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“…In this way artefacts introduced by artificial truncation of the domain are avoided. Equation (5) is discretised and solved by extending the finite-difference numerical method proposed by Coco et al (2014) for Cauchy-Navier equations to thermo-poroelasticity equations.…”

Section: A Coco Et Al: Numerical Models For Ground Deformation and mentioning

confidence: 99%

“…The finite-difference method presented by Coco and Russo (2013) is applied to solve the problem Eq. (7) on an infinite domain, using the coordinate transformation method (Coco et al, 2014).…”

Section: A Coco Et Al: Numerical Models For Ground Deformation and mentioning

confidence: 99%

Numerical models for ground deformation and gravity changes during volcanic unrest: simulating the hydrothermal system dynamics of a restless caldera

et al. 2016

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Abstract. Ground deformation and gravity changes in restless calderas during periods of unrest can signal an impending eruption and thus must be correctly interpreted for hazard evaluation. It is critical to differentiate variation of geophysical observables related to volume and pressure changes induced by magma migration from shallow hydrothermal activity associated with hot fluids of magmatic origin rising from depth. In this paper we present a numerical model to evaluate the thermo-poroelastic response of the hydrothermal system in a caldera setting by simulating pore pressure and thermal expansion associated with deep injection of hot fluids (water and carbon dioxide). Hydrothermal fluid circulation is simulated using TOUGH2, a multicomponent multiphase simulator of fluid flows in porous media. Changes in pore pressure and temperature are then evaluated and fed into a thermo-poroelastic model (one-way coupling), which is based on a finite-difference numerical method designed for axi-symmetric problems in unbounded domains.Informed by constraints available for the Campi Flegrei caldera (Italy), a series of simulations assess the influence of fluid injection rates and mechanical properties on the hydrothermal system, uplift and gravity. Heterogeneities in hydrological and mechanical properties associated with the presence of ring faults are a key determinant of the fluid flow pattern and consequently the geophysical observables. Peaks (in absolute value) of uplift and gravity change profiles computed at the ground surface are located close to injection points (namely at the centre of the model and fault areas).Temporal evolution of the ground deformation indicates that the contribution of thermal effects to the total uplift is almost negligible with respect to the pore pressure contribution during the first years of the unrest, but increases in time and becomes dominant after a long period of the simulation. After a transient increase over the first years of unrest, gravity changes become negative and decrease monotonically towards a steady-state value.Since the physics of the investigated hydrothermal system is similar to any fluid-filled reservoir, such as oil fields or CO 2 reservoirs produced by sequestration, the generic formulation of the model will allow it to be employed in monitoring and interpretation of deformation and gravity data associated with other geophysical hazards that pose a risk to human activity.

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“…13, and the respective discrete right-hand side is a grid function g h ∈ S(∂Ω h ) such that g h (G) = g(B) according to the Eq. (13). With this notation, we can write the linear system on the grid with spatial step h in the following compact form:…”

Section: Multigrid Componentsmentioning

confidence: 99%

“…Elliptic equations with discontinuous coefficients arise from the mathematical modelling of a large number of reallife applications. Examples include the steady-state solution of diffusion problems, for instance in the context of solidification processes of materials with different diffusion coefficients across a complex interface [64]; the Poisson equation arising from the projection method for incompressible two-phase fluids with different physical characteristics [59]; the study of electrostatic phenomena such as those encountered in the simulation of biomolecules' electric potential [42], and many more [40,13]. All these problems may be characterized by complex moving interfaces across which the jump of the solution and its flux must be prescribed for well-posedness.…”

Section: Introductionmentioning

confidence: 99%

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

Coco¹,

Russo²

2018

Journal of Computational Physics

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In this paper we propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients (with general non homogeneous jumps in the solution and its gradient) in 2D and 3D. The method consists of a finite-difference method on a Cartesian grid in which complex geometries (boundaries and interfaces) are embedded, and is second order accurate in the solution and the gradient itself. In order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface, two numerical values are assigned on grid points that are close to the interface: a real value, that represents the numerical solution on that grid point, and a ghost value, that represents the numerical solution extrapolated from the other side of the interface, obtained by enforcing the assigned non-homogeneous jump conditions on the solution and its flux. The method is also extended to the case of matrix coefficient. The linear system arising from the discretization is solved by an efficient multigrid approach. Unlike the 1D case, grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient. A proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor, comparable with the one obtained by Local Fourier Analysis for rectangular domains. The method is robust enough to handle large jump in the coefficients: order of accuracy, monotonicity of the errors and good convergence factor are maintained by the scheme.

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A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

Cited by 26 publications

References 37 publications

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

Numerical models for ground deformation and gravity changes during volcanic unrest: simulating the hydrothermal system dynamics of a restless caldera

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

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