A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

In this paper,a new finite element and finite difference(FE-FD)method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,the standard P,FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite,while near the interface,the maximum principle preserving immersed interface discretization is applied.In the time discretization,a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate.Correction terms are needed when the interface crosses grid lines.The moving interface is represented by the zero level set of a Lipschitz continuous function.Numerical experiments presented in this paper confirm second orderconvergence.

Decay estimates of discretized Green's functions for Schrdinger type operators

For a sparse non-singular matrix A, generally A- 1 is a dense matrix. However, for a class of matrices, A-1 can be a matrix with off-diagonal decay properties, i.e., ｜Aij^-1｜ decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green＇s functions for SchrSdinger type operators. We provide decay estimates for discretized Green＇s functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schr6dinger type operators.

A tight sandstone multi-physical hydraulic fractures simulator study and its field application

During the past years,the recovery of unconventional gas formation has attracted lots of attention and achieved huge success.To produce gas from the low-permeability unconventional formations,hydraulic fracturing technology is essential and critical.In this paper,we present the development of a three-dimensional thermalhydraulic-mechanical numerical simulator for the simulation of hydraulic fracturing operations in tight sandstone reservoirs.Our simulator is based on integrated finite difference(IFD)method.In this method,the simulation domain is subdivided into sub domains and the governing equations are integrated over a sub domain with flux terms expressed as an integral over the sub domain boundary using the divergence theorem.Our simulator conducts coupled thermal-hydraulic-mechanical simulation of the initiation and extension of hydraulic fractures.It also calculates the mass/heat transport of injected hydraulic fluids as well as proppants.Our simulator is able to handle anisotropic formations with multiple layers.Our simulator has been validated by comparing with an analytical solution as well as Ribeiro and Sharma model.Our model can simulate fracture spacing effect on fracture profile when combining IFD with Discontinuous Displacement Method(DDM).