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Radial basis functions are typically used when discretization sche-mesrequire inhomogeneous node distributions. While spawning from a desire tointerpolate functions on a random set of nodes, they have found successfulapplications in solving many types of differential equations. However, theweights of the interpolated solution, used in the linear superposition of basisfunctions to interpolate the solution, and the actual value of the solution arecompletely different. In fact, these weights mix the value of the solution withthe geometrical location of the nodes used to discretize the equation. In thispaper, we used nodal radial basis functions, which are interpolants of theimpulse function at each node inside the domain. This transformation allows tosolve a linear hyperbolic partial differential equation using series expansionrather than the explicit computation of a matrix inverse. This transformationeffectively yields an implicit solver which only requires the multiplication ofvectors with matrices. Because the solver requires neither matrix inverse normatrix-matrix products, this approach is numerically more stable and reducesthe error by at least two orders of magnitude, compared to other solvers usingradial basis functions directly. Further, boundary conditions are integrateddirectly inside the solver, at no extra cost. The method is naturallyconservative, keeping the error virtually constant throughout the computation.

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