Volume 27, Issue 2
Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions

K. Monysekar & Y. V. S. S. Sanyasiraju

Commun. Comput. Phys., 27 (2020), pp. 460-479.

Published online: 2019-12

Preview Purchase PDF 28 1250
Export citation
  • Abstract

RBF based grid-free scheme with PDE centres is experimented in this work for solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemes can give converged solutions for the entire range of diffusivity wherein sharp layers are expected in the solutions and accurate computation of these layers is a big challenge for most of the numerical schemes. Radial Basis Function (RBF) based Local Hermitian Interpolation (LHI) with PDE centres is one such integration scheme which has some built in upwind effect and hence may be a good solver for the convection dominated problems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based scheme with PDE centres, which was initially used to solve some time dependent problems in [10]. RBF based numerical schemes are one type of grid free numerical schemes based on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a variety of steady benchmark problems with continuous and discontinuous boundary data and validated against the corresponding exact solutions. Comparisons of the solutions of the convective dominant benchmark problems show that the upwind biasing either in source centres or PDE centres gives convergent solutions consistently and is stable without any oscillations especially for problems with discontinuities in the boundary conditions. It is observed that the accuracy of the solutions is better than the solutions of other standard integration schemes particularly when the computations are carried out with fewer centers.

  • Keywords

Grid-free scheme, radial basis function, convection-diffusion, multi-quadric, PDE points, upwind.

  • AMS Subject Headings

76D99, 65N06, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

k.monys@gmail.com (K. Monysekar)

sryedida@iitm.ac.in (Y. V. S. S. Sanyasiraju)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-460, author = {Monysekar , K. and Sanyasiraju , Y. V. S. S. }, title = {Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions}, journal = {Communications in Computational Physics}, year = {2019}, volume = {27}, number = {2}, pages = {460--479}, abstract = {

RBF based grid-free scheme with PDE centres is experimented in this work for solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemes can give converged solutions for the entire range of diffusivity wherein sharp layers are expected in the solutions and accurate computation of these layers is a big challenge for most of the numerical schemes. Radial Basis Function (RBF) based Local Hermitian Interpolation (LHI) with PDE centres is one such integration scheme which has some built in upwind effect and hence may be a good solver for the convection dominated problems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based scheme with PDE centres, which was initially used to solve some time dependent problems in [10]. RBF based numerical schemes are one type of grid free numerical schemes based on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a variety of steady benchmark problems with continuous and discontinuous boundary data and validated against the corresponding exact solutions. Comparisons of the solutions of the convective dominant benchmark problems show that the upwind biasing either in source centres or PDE centres gives convergent solutions consistently and is stable without any oscillations especially for problems with discontinuities in the boundary conditions. It is observed that the accuracy of the solutions is better than the solutions of other standard integration schemes particularly when the computations are carried out with fewer centers.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0054}, url = {http://global-sci.org/intro/article_detail/cicp/13454.html} }
TY - JOUR T1 - Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions AU - Monysekar , K. AU - Sanyasiraju , Y. V. S. S. JO - Communications in Computational Physics VL - 2 SP - 460 EP - 479 PY - 2019 DA - 2019/12 SN - 27 DO - http://dor.org/10.4208/cicp.OA-2018-0054 UR - https://global-sci.org/intro/article_detail/cicp/13454.html KW - Grid-free scheme, radial basis function, convection-diffusion, multi-quadric, PDE points, upwind. AB -

RBF based grid-free scheme with PDE centres is experimented in this work for solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemes can give converged solutions for the entire range of diffusivity wherein sharp layers are expected in the solutions and accurate computation of these layers is a big challenge for most of the numerical schemes. Radial Basis Function (RBF) based Local Hermitian Interpolation (LHI) with PDE centres is one such integration scheme which has some built in upwind effect and hence may be a good solver for the convection dominated problems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based scheme with PDE centres, which was initially used to solve some time dependent problems in [10]. RBF based numerical schemes are one type of grid free numerical schemes based on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a variety of steady benchmark problems with continuous and discontinuous boundary data and validated against the corresponding exact solutions. Comparisons of the solutions of the convective dominant benchmark problems show that the upwind biasing either in source centres or PDE centres gives convergent solutions consistently and is stable without any oscillations especially for problems with discontinuities in the boundary conditions. It is observed that the accuracy of the solutions is better than the solutions of other standard integration schemes particularly when the computations are carried out with fewer centers.

K. Monysekar & Y. V. S. S. Sanyasiraju. (2019). Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions. Communications in Computational Physics. 27 (2). 460-479. doi:10.4208/cicp.OA-2018-0054
Copy to clipboard
The citation has been copied to your clipboard