In this paper, we summarize a methodology based on multidimensional finite differencing (MDFD) for the numerical solution of partial differential equations. Although the general methodology is not new, we present some original modifications that use Cartesian-based gridding. In theory, this methodology is applicable for any type of boundary conditions, linear or nonlinear, steady state or unsteady state. The two most important characteristics of the methodology are the simplicity of the formulation and the fact that no distortion or refinement of the Cartesian grid is necessary to accommodate a non-Cartesian (irregular) boundary. For example cases, Neumann and Dirichlet conditions are applied in the numerical solution of the unsteady-state heat-conduction equation.

This paper has three primary benefits. First, we provide a classification system for MDFD methods. Second, we present a new MDFD method involving the combination of interpolation through nodes and regression through boundary conditions. Finally, we present cases where MDFD fails to provide an acceptable result, and other cases where super accuracy occurs.

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