Accurate numerical method to solve flux distribution of Poisson’s equation

This paper proposes an accurate numerical method, the direct flux method (DFM), to solve fluxes directly for Poisson’s equation. In DFM, fluxes are the variables to be solved in the system equations, where a flux is defined as an integral of the flux density across a certain finite-sized cross section. The system equation of the DFM is derived from two equations: an integral form of Poisson’s equation obtained by using Gauss’s divergence theorem and an integral form of the rotation-free nature of any scalar field from Stokes’ theorem. In the numerical approach, no discretization error arises from Gauss’s divergence theorem because it is represented as a sum of fluxes. Therefore, the discretization error is caused only by the integral form of the rotation-free nature. From the comparison between DFM, the finite difference method (FDM), and the finite volume method (FVM), we show that the accuracy of DFM is superior to that of FDM and FVM. However, DFM generally requires larger computational resources than other methods because the number of equations in DFM is more than that in other methods. To overcome this drawback, we also propose a faster algorithm than DFM, called FastDFM, which can reduce the number of equations without changing the accuracy. Hence, the proposed FastDFM produces results with the same accuracy as the DFM and with computation time almost the same as that of FDM and FVM.