scholarly journals On the rate of convergence of finite difference schemes on nonuniform grids

1985 ◽  
Vol 26 (3) ◽  
pp. 247-256 ◽  
Author(s):  
Frank De Hoog ◽  
David Jackett
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Numerical Approximation ◽  
Truncation Error ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Local Truncation Error ◽  
Point Boundary ◽  
Nonuniform Grids ◽  
Higher Dimensional

AbstractFinite difference schemes for some two point boundary value problems are analysed. It is found that for schemes defined on nonuniform grids, the order of the local truncation error does not fully reflect the rate of convergence of the numerical approximation obtained. Numerical results are presented that indicate that this is also the case for higher dimensional problems.

scholarly journals SIXTH-ORDER ACCURATE FINITE DIFFERENCE SCHEMES FOR THE HELMHOLTZ EQUATION

2006 ◽  
Vol 14 (03) ◽  
pp. 339-351 ◽  
Author(s):  
I. SINGER ◽  
E. TURKEL
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Truncation Error ◽  
Model Problem ◽  
Difference Schemes ◽  
Sixth Order ◽  
Finite Difference Schemes ◽  
Local Truncation Error ◽  
Two Dimensional ◽  
Truncation Order

We develop and analyze finite difference schemes for the two-dimensional Helmholtz equation. The schemes which are based on nine-point approximation have a sixth-order accurate local truncation order. The schemes are compared with the standard five-point pointwise representation, which has second-order accurate local truncation error and a nine-point fourth-order local truncation error scheme based on a Padé approximation. Numerical results are presented for a model problem.

scholarly journals Rate of convergence and error estimates for finite-difference schemes of solving linear ill-posed Cauchy problems of the second order

2017 ◽  
pp. 322-347
Author(s):  
М.М. Кокурин
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Error Estimates ◽  
Sufficient Conditions ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Finite Difference Schemes ◽  
Cauchy Problems ◽  
Ill Posed

Изучаются конечно-разностные схемы решения некорректных задач Коши для линейного дифференциально-операторного уравнения второго порядка в банаховом пространстве. Получены равномерные по времени оценки скорости сходимости и погрешности этих схем при наложении на искомое решение условия истокопредставимости. Найдены близкие друг к другу необходимые и достаточные условия в терминах показателя истокопредставимости для сходимости класса схем со степенной скоростью относительно шага дискретизации. Построены и изучены схемы полной дискретизации некорректных задач Коши второго порядка, сочетающие полудискретизацию по времени с дискретной аппроксимацией пространства и оператора. Finite-difference schemes of solving ill-posed Cauchy problems for linear second-order differential operator equations in Banach spaces are considered. Several time-uniform rate of convergence and error estimates are obtained for the considered schemes under the assumption that the sought solution satisfies the sourcewise condition. Necessary and sufficient conditions are found in terms of sourcewise index for a class of schemes with the power convergence rate with respect to the discretization step. A number of full discretization schemes for second-order ill-posed Cauchy problems are proposed on the basis of combining the half-discretization in time with the discrete approximation of the spaces and the operators.

Accuracy of Higher-Order Finite Difference Schemes on Nonuniform Grids

AIAA Journal ◽  
2003 ◽  
Vol 41 (8) ◽  
pp. 1609-1611 ◽  
Author(s):  
Yongmann M. Chung ◽  
Paul G. Tucker
Keyword(s):  
Finite Difference ◽  
Higher Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Nonuniform Grids

scholarly journals Solving nonlinear two point boundary value problems using exponential finite difference method

2016 ◽  
Vol 34 (1) ◽  
pp. 33-44
Author(s):  
Pramod K. Pandey
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Truncation Error ◽  
Boundary Value ◽  
Local Truncation Error ◽  
Point Boundary ◽  
Established Order ◽  
Good Agreement

In this article, we present exponential finite difference scheme for solving nonlinear two point boundary value problems with Dirichlet's boundary conditions . The local truncation error and under appropriate condition we have discussed the convergence of the proposed method. Numerical experiments demonstrate the use and computational efficiency of the method. Numerical results show that this method is at least fourth order accurate, which is good agreement with the theoretically established order of the method.

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