The general solution of some finite difference equations analogous to the wave equation

1971 ◽  
Vol 7 (1) ◽  
pp. 122-122
Author(s):  
M. A. McKiernan
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
General Solution ◽  
Difference Equations ◽  
Finite Difference Equations

scholarly journals Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting

2020 ◽  
Vol 7 (1) ◽  
pp. 48-55 ◽  
Author(s):  
Bolat Duissenbekov ◽  
Abduhalyk Tokmuratov ◽  
Nurlan Zhangabay ◽  
Zhenis Orazbayev ◽  
Baisbay Yerimbetov ◽  
...  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Constant Load ◽  
Time Interval ◽  
Test Case ◽  
Algebraic Equations ◽  
Nonlinear Properties ◽  
Finite Difference Equations ◽  
Concrete Shells

AbstractThe study solves a system of finite difference equations for flexible shallow concrete shells while taking into account the nonlinear deformations. All stiffness properties of the shell are taken as variables, i.e., stiffness surface and through-thickness stiffness. Differential equations under consideration were evaluated in the form of algebraic equations with the finite element method. For a reinforced shell, a system of 98 equations on a 8×8 grid was established, which was next solved with the approximation method from the nonlinear plasticity theory. A test case involved computing a 1×1 shallow shell taking into account the nonlinear properties of concrete. With nonlinear equations for the concrete creep taken as constitutive, equations for the quasi-static shell motion under constant load were derived. The resultant equations were written in a differential form and the problem of solving these differential equations was then reduced to the solving of the Cauchy problem. The numerical solution to this problem allows describing the stress-strain state of the shell at each point of the shell grid within a specified time interval.

Tests of the Stability and Time-Step Sensitivity of Semi-Implicit Reservoir Stimulation Techniques

1972 ◽  
Vol 12 (03) ◽  
pp. 253-266 ◽  
Author(s):  
James S. Nolen ◽  
D.W. Berry
Keyword(s):  
Finite Difference ◽  
Difference Equations ◽  
Implicit Method ◽  
Significant Advance ◽  
Relative Permeabilities ◽  
Time Step ◽  
Finite Difference Equations ◽  
Nonlinear Form ◽  
Implicit Finite Difference ◽  
Excellent Stability

Abstract A reservoir simulation technique that employs semi-implicit approximations to relative permeabilities exhibits excellent stability and permeabilities exhibits excellent stability and convergence characteristics when applied to water- or gas-coning problems. Recent workers in this area have made a simplifying assumption in order to linearize the flow terms of the semi-implicit finite-difference equations. This paper describes a method of solving efficiently paper describes a method of solving efficiently the nonlinear form of the equations and demonstrates that time-step sensitivity is reduced by iterating on the nonlinear terms. In addition, it addresses the problem of allocating a well's production among multiple grid blocks. Example problems include both water-coning and gas-percolation applications. Introduction Multiphase reservoir simulators traditionally have employed finite-difference approximations in which relative permeabilities are evaluated explicitly at the beginning of each time step. Simulators of this type are capable of handling many reservoir studies in a perfectly satisfactory fashion, but they are incapable of solving economically problems characterized by high flow velocities. Included in this category are the studies of such phenomena as well coning and gas percolation. The difficulty in such problems is a stability limitation imposed by the use of explicit relative permeabilities. In an attempt to overcome this permeabilities. In an attempt to overcome this limitation, Blair and Weinaug developed a simulator that employed implicitly evaluated relative permeabilities. The increased stability of their permeabilities. The increased stability of their equations allowed the use of time steps much larger than previously possible, but this was counteracted by an increase in the computational work per time step and an increased difficulty in the iterative solution of the difference equations. While the net result was a significant advance in the solution of coning problems, improvements still were needed to increase the dependability and decrease the cost of obtaining solutions for such problems. More recently, two papers were published describing a method that employs semi-implicit relative permeabilities. This method is greatly superior to the fully implicit method, both in computational effort and maximum time-step size. In developing this method, the previous workers made a simplifying assumption to obtain linear finite-difference equations. We have developed a reservoir simulator based on the nonlinear form of the semi-implicit finite-difference equations. This paper describes the techniques used in the simulator and presents the results of some tests conducted with it. These include time-step sensitivity studies and tests of alternate production allocation methods. Some of these tests compare the nonlinear form of the semi-implicit method with the linear form. SPEJ P. 253

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