A fully implicit method using nodal radial basis functions to solve the linear advection equation

TL;DR

This paper introduces a fully implicit radial basis function method using nodal interpolants to solve the linear advection equation, offering improved stability, accuracy, and direct boundary condition integration.

Contribution

The paper presents a novel implicit solver based on nodal radial basis functions that avoids matrix inversion, enhances numerical stability, and simplifies boundary condition implementation.

Findings

Reduces error by at least two orders of magnitude compared to existing methods.

Eliminates the need for matrix inverse and matrix-matrix products.

Maintains constant error and conserves the solution throughout computation.

Abstract

Radial basis functions are typically used when discretization sche-mes 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 other solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is naturally conservative, keeping the error virtually constant throughout the computation.