The Concept of the Differential Approximation of a Difference Scheme

1983 ◽  
pp. 8-37
Author(s):  
Yu. I. Shokin
Keyword(s):  
Difference Scheme ◽  
Differential Approximation

scholarly journals INVESTIGATION OF A COST‐EFFECTIVE FINITE DIFFERENCE SCHEME FOR THE TRANSPORT EQUATION

2003 ◽  
Vol 8 (3) ◽  
pp. 247-258
Author(s):  
D. G. Slugin ◽  
A. V. Popov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Transport Equation ◽  
Finite Difference Scheme ◽  
Three Dimensional ◽  
Cost Effective ◽  
Differential Approximation ◽  
Fully Discrete ◽  
Implicit Finite Difference Scheme ◽  
Implicit Finite Difference

The transport equation for three‐dimensional flow of a viscous gas is considered. An implicit finite difference scheme is constructed for approximating the transport equation. The error estimation is proved. The main part of the analysis is done for the first differential approximation of the proposed finite difference scheme, but the results are also valid in the fully discrete case.

High order modified differential equation of the Beam–Warming method, II. The dissipative features

2020 ◽  
Vol 35 (3) ◽  
pp. 175-185
Author(s):  
Yurii Shokin ◽  
Ireneusz Winnicki ◽  
Janusz Jasinski ◽  
Slawomir Pietrek
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wind Speed ◽  
Difference Scheme ◽  
Advection Equation ◽  
Explicit Difference Scheme ◽  
Eighth Order ◽  
Differential Approximation ◽  
Partial Differential ◽  
Linear Advection Equation

AbstractThis paper is a continuation of [38]. The analysis of the modified partial differential equation (MDE) of the constant-wind-speed linear advection equation explicit difference scheme up to the eighth-order derivatives is presented. In this paper the authors focus on the dissipative features of the Beam–Warming scheme. The modified partial differential equation is presented in the so-called Π-form of the first differential approximation. The most important part of this form includes the coefficients μ (p) at the space derivatives. Analysis of these coefficients provides indications of the nature of the dissipative errors. A fragment of the stencil for determining the modified differential equation for the Beam–Warming scheme is included. The derived and presented coefficients μ (p) as well as the analysis of the dissipative features of this scheme on the basis of these coefficients have not been published so far.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

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