scholarly journals Infinite Schrödinger networks

2021 ◽  
Vol 31 (4) ◽  
pp. 640-650
Author(s):  
N. Nathiya ◽  
C. Amulya Smyrna
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Laplace Operator ◽  
Canonical Representation ◽  
Poisson Equations ◽  
Superharmonic Functions ◽  
The Poisson Equation ◽  
Infinite Network ◽  
Discrete Potential Theory

Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

Simulation of a Gas-Dominated, Two-Phase Geothermal Reservoir

1980 ◽  
Vol 20 (01) ◽  
pp. 52-58 ◽  
Author(s):  
G.A. Zyvoloski ◽  
M.J. O'Sullivan
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Gas Content ◽  
Natural State ◽  
Petroleum Reservoir ◽  
Partial Differential ◽  
Geothermal Reservoirs ◽  
Noncondensable Gas ◽  
Newton Raphson

Abstract The basic equations governing the behavior of a two-phase mixture of carbon dioxide (CO2) and water are discussed. A Newton-Raphson scheme, based on the alternating direction implicit (ADI) method for multidimensional problems, is used to solve the nonlinear finite difference approximation of the governing nonlinear system of partial differential equations. Sample calculations showing the behavior of hypothetical reservoirs with varying CO2 contents are presented. Introduction Geothermal reservoirs often contain an appreciable amount of noncondensable gases that have major effects on the behavior of the reservoir in both its natural state and under exploitation. In its natural state, the partial pressure of the noncondensable gas causes the reservoir to boil at a lower temperature than does a pure water field. Under exploitation the presence of CO2 or hydrogen sulfide (the most presence of CO2 or hydrogen sulfide (the most common gases in geothermal fields) dominates the transport and thermodynamical characteristics of the flow. It already has been shown that for lumped parameter models of geothermal reservoirs, small parameter models of geothermal reservoirs, small differences in the CO2 content of the reservoir cause major changes in the pressure, enthalpy, and gas content of the discharge fluid. Because of the importance of the gas content in influencing the design of flash turbines and other geothermal energy conversion systems, it is essential to include the effects of noncondensable gases in simulations of gassy geothermal reservoirs.Most of the procedures developed for simulating the behavior of geothermal reservoirs have features that are not modified easily to include the effects of a noncondensable gas. These difficulties arise either from the numerical procedures used or from the methods of treatment of the thermodynamics of the system. The work of Pritchett et al. uses the fluid (mixture) density and internal energy as primary dependent variables and interpolation between tabular values for determining thermodynamic properties. With the addition of a noncondensable properties. With the addition of a noncondensable gas, the use of density and internal energy (together with one other unknown) leads to an indirect iterative calculation of the other fluid properties such as temperature and pressure. Even when the noncondensable gas is not present, the use of pressure as one of the unknowns appears to be desirable.The procedure developed by Faust and Mercer uses the same unknowns used here, namely pressure and mixture enthalpy, but their procedure for solving the finite difference equations involved requires some modification to allow the introduction of the extra unknown arising from the presence of a noncondensable gas. The numerical methods used by Mercer and Faust, Thomas, and Coats all are based on the earlier work of Price and Coats. Basically the problem involves implicit nonlinear finite difference approximations of the governing partial differential equations that are solved using the partial differential equations that are solved using the Newton-Raphson method. For multidimensional problems the matrices that arise in the problems the matrices that arise in the Newton-Raphson procedure are sparse but have a large bandwidth. Careful ordering of the equations enables the bandwidth to be reduced significantly. Still greater numerical efficiency is possible by using the alternating direction procedure combined with the lagging of corrections to the permeability terms in the Newton-Raphson process, The application of this ADI procedure to petroleum reservoir problems is restricted severely by petroleum reservoir problems is restricted severely by stability limits on the time step, but compressibility and thermal expansion effects in geothermal problems tend to stabilize the scheme. problems tend to stabilize the scheme. JPT P. 52

scholarly journals CMMSE: A technique for generating adapted discretizations to solve partial differential equations with the generalized finite difference method

2021 ◽  
Author(s):  
Augusto César Ferreira ◽  
Miguel Ureña ◽  
HIGINIO RAMOS
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Meshless Method ◽  
Computational Cost ◽  
Partial Differential

The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strategy allows using a smaller number of points and a lower computational cost to achieve the same accuracy that would be obtained with a regular discretization.

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