scholarly journals Numerical analysis of one-dimensional convective diffusion equation by weighted finite difference equations.

1985 ◽  
pp. 97-104
Author(s):  
Masamichi KANOH ◽  
Toshihiko UEDA
Keyword(s):  
Numerical Analysis ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Equations ◽  
Convective Diffusion ◽  
One Dimensional ◽  
Finite Difference Equations ◽  
Convective Diffusion Equation

Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation Galerkin Method

2006 ◽  
Author(s):  
James Geer ◽  
John Fillo
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Galerkin Method ◽  
Difference Equations ◽  
Space Variable ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hybrid Technique ◽  
Finite Difference Equations ◽  
Hybrid Equations

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods, based on the parameter ρ = Δt / (Δx)2, with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include: 1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; 2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and 3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

scholarly journals On some newN-independent-variable discrete inequalities of the Gronwall type

1986 ◽  
Vol 9 (3) ◽  
pp. 485-495 ◽  
Author(s):  
En Hao Yang
Keyword(s):  
Numerical Analysis ◽  
Finite Difference ◽  
Difference Equations ◽  
Independent Variables ◽  
Finite Difference Equations ◽  
Discrete Inequalities ◽  
Independent Variable

In this paper we shall establish some new discrete inequalities of the Gronwall type inN-independent variables. They will have many applications for finite difference equations involving several independent variables and for numerical analysis. Their consequence for the case ofN=3, generalizes all of the known theorems obtained by Pachpatte and Singare in [1]. An example, to which those results established in [1] are inapplicable, is given here to convey the usefulness of the results obtained.

Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation–Galerkin Method

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
James Geer ◽  
John Fillo
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Galerkin Method ◽  
Difference Equations ◽  
Space Variable ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hybrid Technique ◽  
Finite Difference Equations ◽  
Hybrid Equations

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods based on the parameter ρ=Δt∕(Δx)2 with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include the (1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; (2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and (3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

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.

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