A time-marching MFS scheme for heat conduction problems

In this work we consider the numerical solution of a heat conduction problem for a material with non-constant properties. By approximating the time derivative of the solution through a finite difference, the transient equation is transformed into a sequence of inhomogeneous Helmholtz-type equations. The corresponding elliptic boundary value problems are then solved numerically by a meshfree method using fundamental solutions of the Helmholtz equation as shape functions. Convergence and stability of the method are addressed. Some of the advantages of this scheme are the absence of domain or boundary discretizations and/or integrations. Also, no auxiliary analytical or numerical methods are required for the derivation of the particular solution of the inhomogeneous elliptic problems. Numerical simulations for 2D domains are presented. Smooth and non-smooth boundary data will be considered.