STABILITY OF THE FOUR-POINT FINITE DIFFERENCE SCHEME FOR LINEAR PARABOLIC EQUATIONS WITH FIRST DERIVATIVES AND WITH VARIABLE COEFFICIENTS

1976 ◽  
Vol 9 (2) ◽  
Author(s):  
Jacek Wojtowicz
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Parabolic Equations ◽  
Finite Difference Scheme ◽  
Variable Coefficients ◽  
Linear Parabolic Equations

scholarly journals Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

2006 ◽  
Vol 6 (2) ◽  
pp. 154-177 ◽  
Author(s):  
E. Emmrich ◽  
R.D. Grigorieff
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Elliptic Boundary ◽  
Special Kind ◽  
Variable Coefficients ◽  
Second Order ◽  
Nonuniform Grids ◽  
Close Relationship

AbstractIn this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given.

scholarly journals A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Changkai Chen ◽  
Xiaohua Zhang ◽  
Zhang Liu ◽  
Yage Zhang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Parabolic Equations ◽  
Finite Difference Scheme ◽  
Integration Method ◽  
Computational Accuracy ◽  
Precise Integration ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Taylor Approximation

AbstractThis paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a $(4,4)$(4,4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the $(4,4)$(4,4)-Padé approximation.

Convergence Of a Finite Difference Scheme for the Third Boundary-Value Problem for an Elliptic Equation with Variable Coefficients

2001 ◽  
Vol 1 (4) ◽  
pp. 356-366 ◽  
Author(s):  
Boško S. Jovanović ◽  
Branislav Z. Popović
Keyword(s):  
Elliptic Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Boundary Value ◽  
Variable Coefficients ◽  
Convergence Rate Estimate ◽  
The Third ◽  
Third Boundary Value Problem

Abstract In this paper we study the convergence of a finite difference scheme that approximates the third boundary-value problem for an elliptic equation with variable coefficients on a unit square. An ”almost” second-order convergence rate estimate (with additional logarithmic multiplier) is obtained.

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