scholarly journals Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Peyman Hessari ◽  
Sang Dong Kim ◽  
Byeong-Chun Shin
Keyword(s):  
Collocation Method ◽  
Source Term ◽  
Spectral Element ◽  
Rectangular Domain ◽  
Interface Problems ◽  
Interface Problem ◽  
Spectral Collocation ◽  
Elliptic Interface Problem ◽  
Spectral Convergence ◽  
Singular Source

The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.

scholarly journals A SPECTRAL COLLOCATION APPROXIMATION FOR THE RADIAL-INFALL OF A COMPACT OBJECT INTO A SCHWARZSCHILD BLACK HOLE

2009 ◽  
Vol 20 (11) ◽  
pp. 1827-1848 ◽  
Author(s):  
JAE-HUN JUNG ◽  
GAURAV KHANNA ◽  
IAN NAGLE
Keyword(s):  
Black Hole ◽  
Collocation Method ◽  
Projection Method ◽  
Difference Method ◽  
Point Particle ◽  
Spectral Collocation Method ◽  
Source Terms ◽  
Schwarzschild Black Hole ◽  
Spectral Collocation ◽  
Singular Source

The inhomogeneous Zerilli equation is solved in time-domain numerically with the Chebyshev spectral collocation method to investigate a radial-infall of the point particle towards a Schwarzschild black hole. Singular source terms due to the point particle appear in the equation in the form of the Dirac δ-function and its derivative. For the approximation of singular source terms, we use the direct derivative projection method proposed in Ref. 9 without any regularization. The gravitational waveforms are evaluated as a function of time. We compare the results of the spectral collocation method with those of the explicit second-order central-difference method. The numerical results show that the spectral collocation approximation with the direct projection method is accurate and converges rapidly when compared with the finite-difference method.

A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations

2020 ◽  
Vol 21 (7-8) ◽  
pp. 683-691
Author(s):  
Phumlani G. Dlamini ◽  
Vusi M. Magagula
Keyword(s):  
Collocation Method ◽  
Three Dimensional ◽  
Linearization Method ◽  
High Accuracy ◽  
Time Variable ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Burgers Equations ◽  
Grid Points ◽  
Linearization Techniques

AbstractIn this paper, we introduce the multi-variate spectral quasi-linearization method which is an extension of the previously reported bivariate spectral quasi-linearization method. The method is a combination of quasi-linearization techniques and the spectral collocation method to solve three-dimensional partial differential equations. We test its applicability on the (2 + 1) dimensional Burgers’ equations. We apply the spectral collocation method to discretize both space variables as well as the time variable. This results in high accuracy in both space and time. Numerical results are compared with known exact solutions as well as results from other papers to confirm the accuracy and efficiency of the method. The results show that the method produces highly accurate solutions and is very efficient for (2 + 1) dimensional PDEs. The efficiency is due to the fact that only few grid points are required to archive high accuracy. The results are portrayed in tables and graphs.

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