Volume 20, Issue 3
A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation

Int. J. Numer. Anal. Mod., 20 (2023), pp. 437-458.

Published online: 2023-03

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• Abstract

Radial basis functions are typically used when discretization schemes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in solving many types of differential equations. However, the weights of the interpolated solution, used in the linear superposition of basis functions to interpolate the solution, and the actual value of the solution are completely different. In fact, these weights mix the value of the solution with the geometrical location of the nodes used to discretize the equation. In this paper, we used nodal radial basis functions, which are interpolants of the impulse function at each node inside the domain. This transformation allows to solve a linear hyperbolic partial differential equation using series expansion rather than the explicit computation of a matrix inverse. This transformation effectively yields an implicit solver which only requires the multiplication of vectors with matrices. Because the solver requires neither matrix inverse nor matrix-matrix products, this approach is numerically more stable and reduces the error by at least two orders of magnitude, compared to solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is locally conservative, keeping the error virtually constant throughout the computation.

65D12, 65F05, 35F50, 35F46, 65D05, 35L65

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@Article{IJNAM-20-437, author = {Gourdain , P.-A.Evans , M.Hasson , H. R.Young , J. R.West-Abdallah , I. and Adams , M. B.}, title = {A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {3}, pages = {437--458}, abstract = {

Radial basis functions are typically used when discretization schemes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in solving many types of differential equations. However, the weights of the interpolated solution, used in the linear superposition of basis functions to interpolate the solution, and the actual value of the solution are completely different. In fact, these weights mix the value of the solution with the geometrical location of the nodes used to discretize the equation. In this paper, we used nodal radial basis functions, which are interpolants of the impulse function at each node inside the domain. This transformation allows to solve a linear hyperbolic partial differential equation using series expansion rather than the explicit computation of a matrix inverse. This transformation effectively yields an implicit solver which only requires the multiplication of vectors with matrices. Because the solver requires neither matrix inverse nor matrix-matrix products, this approach is numerically more stable and reduces the error by at least two orders of magnitude, compared to solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is locally conservative, keeping the error virtually constant throughout the computation.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1018}, url = {http://global-sci.org/intro/article_detail/ijnam/21541.html} }
TY - JOUR T1 - A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation AU - Gourdain , P.-A. AU - Evans , M. AU - Hasson , H. R. AU - Young , J. R. AU - West-Abdallah , I. AU - Adams , M. B. JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 437 EP - 458 PY - 2023 DA - 2023/03 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1018 UR - https://global-sci.org/intro/article_detail/ijnam/21541.html KW - Radial basis functions, implicit scheme, hyperbolic equations. AB -

Radial basis functions are typically used when discretization schemes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in solving many types of differential equations. However, the weights of the interpolated solution, used in the linear superposition of basis functions to interpolate the solution, and the actual value of the solution are completely different. In fact, these weights mix the value of the solution with the geometrical location of the nodes used to discretize the equation. In this paper, we used nodal radial basis functions, which are interpolants of the impulse function at each node inside the domain. This transformation allows to solve a linear hyperbolic partial differential equation using series expansion rather than the explicit computation of a matrix inverse. This transformation effectively yields an implicit solver which only requires the multiplication of vectors with matrices. Because the solver requires neither matrix inverse nor matrix-matrix products, this approach is numerically more stable and reduces the error by at least two orders of magnitude, compared to solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is locally conservative, keeping the error virtually constant throughout the computation.

P.-A. Gourdain, M. Evans, H. R. Hasson, J. R. Young, I. West-Abdallah & M. B. Adams. (2023). A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation. International Journal of Numerical Analysis and Modeling. 20 (3). 437-458. doi:10.4208/ijnam2023-1018
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