A good spatial discretisation in the method of lines

Sharaf, Amr A.; Bakodah, H. O.

doi:10.1016/j.amc.2005.01.144

Cited by 9 publications

(6 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Collocation methods for solving the Cauchy problem are constructed on interpolation polynomials whose degrees coincide with the number of collocation points, and the values of the polynomials at these points coincide with the right-hand sides of the differential equation at the calculated points 12,13,17 . Using as the collocation points the set of points of the uniform grid along each formed line: tn,i = tn,0 + iτ ∈ [tn, -m+1, tn,s ], i = -(m -1), -(m -2),…,0,1,…, s, a canonical form of multi-step collocation block methods with the number of reference points m and the number of calculated points 5,14 s:…”

Section: Numerical Realization Of the Methods Of Direct Collocation B...mentioning

confidence: 99%

Parallel implementation of evolutionary partial differential equations by collocation optical-electronic schemes

Bashkov

Дмитриева²,

Huskova³

et al. 2022

Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 2022

View full text Add to dashboard Cite

The problem of parallel solution of partial differential equations with the help of the method of lines that ensures the reduction of the initial problem to the Cauchy problem described by a system of ordinary differential equations is considered. As a basic method, collocated multi-step block difference schemes are proposed. Obtaining a numerical solution in this case becomes possible only with the use of high-performance computing, usually with parallel architecture. This problem becomes especially important in the implementation of mathematical models based on systems of partial differential equations when it comes to the need for discretization of the solution search area, which can be significantly complicated by the geometric configuration of the boundaries. Solution of systems of this order cannot be obtained without involving multiprocessor computers. But by simply increasing the processing power, this problem cannot be solved. Only by combining the advantages of supercomputers and modern numerical simulation methods we can expect a significant improvement in the numerical solution of partial differential equations (PDEs).

show abstract

Section: Numerical Realization Of the Methods Of Direct Collocation B...mentioning

confidence: 99%

Parallel implementation of evolutionary partial differential equations by collocation optical-electronic schemes

Bashkov

Дмитриева²,

Huskova³

et al. 2022

Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 2022

View full text Add to dashboard Cite

show abstract

“…In this paper, a modified method of lines using a new three-point difference [12] is used. The use of this new differences leads to stable schemes with good accuracy.…”

Section: Modified Of the Methods Of Linesmentioning

confidence: 99%

“…The aim of new method is to develop a good formula with high accuracy for the numerical solution of the advection equation using the spatial discretization presented by Sharaf and Bakodah, [12], using 3-points formula, thus the approximate form of the first derivative is ( )…”

Section: Finite Difference Methods With a Good Spatial Discretizationmentioning

confidence: 99%

See 1 more Smart Citation

A Comparative Study of Two Spatial Discretization Schemes for Advection Equation

Bakodah¹

2016

IJMNTA

View full text Add to dashboard Cite

In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both discretization techniques approximate correctly solution of advection equation and compare their accuracy and performance.

show abstract

“…In this paper, modified method of lines using a new three-point difference [3] is used. Use of this new differences leads to stable schemes with good accuracy.…”

Section: Introductionmentioning

confidence: 99%