A Flexible Derivation Approach for the Numerical Solution of Partial Differential Equations

We propose a new method for the numerical solution of boundary value problems associated to partial differential equations. This method is based on standard approximation techniques, like numerical differentiation of univariate functions and curve interpolation, so it can be easily generalized to high-dimensional problems. However, the concrete implementation of this method requires the proper solution of a routing problem for the graph associated with the discretized domain N. This graph routing problem has an immediate solution when N has a grid structure. Instead, when N has a mesh structure or is given by sparse points, it is possible to take advantage of methods for two classical graph routing problems, that is, the Chinese postman problem and the Eulerian path problem. However, it will be shown that these problems do not provide a sufficiently satisfactory solution, hence, further study for non-structured grids is needed. A numerical experiment shows the effectiveness of the proposed method in the case of N with a grid structure.