Abstract
The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be used. Moreover, the numerical values that approximate the functions for the considered domain are obtained. Numerous examples for both equations on irregular 2D domains are exposed to underline the effectiveness and practicality of the method.
Highlights
It is well known that many problems given by non-linear PDEs appear in many fields of science [1,2]
The Eikonal equation is used to describe the behavior of different systems related to control, propagation of waves or environmental phenomena
Θ(ζi2, ηi2) denotes the local truncation error of order 2, as proven in [17,21]. By substituting these in the Eikonal Equation (1) or the stationary heat transfer Equation (2), we obtain a set of non-linear equations as follows: Hp[uq] = 0, p, q = 0, · · ·, N
Summary
It is well known that many problems given by non-linear PDEs appear in many fields of science [1,2]. In [12], a localized boundary-domain integro-differential equation approach is followed, and a wide summary of numerical methods for solving the stationary heat transfer equation is given in [13]. The GFDM is a meshless method, as it solves the PDE at each of the points (nodes) where the domain is discretized. The discretization of the spatial partial derivatives uses a very simple expression (depending only on the distribution of a few nodes, as we explain ), so the treatment of non-linearities is straightforward.
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