The meshless Kernel-based method of lines for parabolic equations

1. Introdu tion. There are plenty of appli ation papers in whi h kernels or radial basis fun tions are su essfully used for solving partial di erential equations by meshless methods. The usage of kernels is typi ally based on spatial interpolation at s attered lo ations, writing the trial fun tions entirely in terms of nodes [2℄. For stationary partial di erential equations, the dis retization an take pointwise analyti derivatives of the trial fun tions to end up with a linear system of equations. This started in [6℄ and was pursued in the following years, in luding a onvergen e theory in [12℄. There are also variations that use weak data, like the Meshless Lo al Petrov Galerkin method [1℄ with a onvergen e theory in [14℄. For the potential equation, there are spe ial kernels that allow the use of trial fun tions that satisfy the di erential equation exa tly [13, 5℄. For time dependent partial di erential equations, meshless kernel based methods were similarly based on a xed spatial interpolation, but now the oe ients are time dependent, and one obtains a system of ordinary di erential equations for these. This is the well knownMethod of Lines, and it turned to be experimentally useful in various ases (see e.g. [16, 7, 4, 15℄). However, a rigid analysis of its behavior seems to be still missing. For the simple ase of the heat equation, this paper provides an analysis of the Method of Lines. To this end, we start with basi s on kernels, then des ribe the Method of Lines and analyze it. Though the Method of Lines needs no expli it CFL ondition, we show how a CFL ondition a ts behind ths s ene. Some numeri al examples are provided as well, and a short se tion showing how to generalize this to mu h more general paraboli equations.