scholarly journals Meshless Finite Difference Operators from Moving Least Squares Interpolation: Applications to PDEs and Convergence Results

PAMM ◽  
2008 ◽  
Vol 8 (1) ◽  
pp. 10847-10848
Author(s):  
Oliver Nowak
Keyword(s):  
Finite Difference ◽  
Least Squares ◽  
Moving Least Squares ◽  
Difference Operators ◽  
Convergence Results ◽  
Moving Least Squares Interpolation

scholarly journals Optimal finite-difference operators for arbitrarily sampled data

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. F39-F51 ◽  
Author(s):  
Erik F. M. Koene ◽  
Johan O. A. Robertsson
Keyword(s):  
Finite Difference ◽  
Cost Function ◽  
Least Squares ◽  
Relative Error ◽  
Acoustic Waves ◽  
Total Error ◽  
Difference Operators ◽  
Relative Cost ◽  
Irregular Grid ◽  
Least Squares Solution

We have developed a general method to obtain the equiripple and the least-squares finite-difference (FD) operator weights to compute arbitrary-order derivatives from arbitrary sample locations. The method is based on the complex-valued Remez exchange algorithm applied to three cost functions: the total error, the relative error, and the group-velocity error. We evaluate the method on three acoustic FD modeling examples. In the first example, we assess the accuracy obtained with the optimal coefficients when propagating acoustic waves through a medium. In the second example, we propagate a wave through an irregular grid. In the final example, we position a source and receiver at arbitrary locations in-between the modeling grid points. In the examples using regular grids, the equiripple solution to the relative cost function performs best. It obtains marginally (4%–10%) better results compared to the second-best option, the least-squares solution for the relative cost function. The least-squares solution for the relative error produced the only stable and accurate results also in the example of modeling on an irregular grid.

Moving Least Squares (MLS) Method for the Nonlinear Hyperbolic Telegraph Equation with Variable Coefficients

2017 ◽  
Vol 14 (03) ◽  
pp. 1750026 ◽  
Author(s):  
A. Mardani ◽  
M. R. Hooshmandasl ◽  
M. M. Hosseini ◽  
M. H. Heydari
Keyword(s):  
Finite Difference ◽  
Least Squares ◽  
Telegraph Equation ◽  
Variable Coefficients ◽  
Moving Least Squares ◽  
Algebraic Equations ◽  
Mesh Structure ◽  
Background Mesh ◽  
Mls Approximation ◽  
Mls Method

Telegraph equation which widely used for modeling many engineering and physical phenomena has considered by some researchers in recent years. In this paper, a numerical scheme based on the moving least squares (MLS) approximation and finite difference method (FDM) is proposed for solving a class of the nonlinear hyperbolic telegraph equation with variable coefficients. In the new developed scheme, we use collocation points and approximate solution of the problem under study by using MLS approximation. The MLS method is a meshless approach and does not need any background mesh structure. A time stepping approach is employed for the first- and second-order time derivatives. The proposed method provides a semi-discrete solutions for the problems under study. In space domain, the MLS approximation and in time domain, the finite difference technique are employed. This method after discretization leads to a linear system of algebraic equations. Some numerical results are given and compared with analytical solutions to demonstrate the validity and efficiency of the proposed technique.

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