A new and rigorous SPN theory – The concluding Part V: Completion of the numerical solution method for GSP3 equations

Abstract

We published a series of four papers on the development of the Generalized SP_N (GSP_N) theory. The theory contains only diffusion equations and composes of different levels of approximation, $GS{P}_N^{\left(K\right)}$. Its highest level $GS{P}_N^{\left(N\right)}$ is equivalent to P_N, while its lowest level $GS{P}_N^{\left(0\right)}$ has the same differential equations as the traditional SP_N but with different interface and boundary conditions. These conditions contain additional tangential derivatives on the surfaces, which poses a serious challenge to traditional methods of numerical calculation. In Part IV of the series, we introduced the generalized transverse integration nodal (GTIN) method to solve the $GS{P}_3^{\left(0\right)}$ equations. Despite the very encouraging numerical results, there was the problem that the base functions could satisfy only part, not all, of the interface and boundary conditions. In this concluding part of the series, we resolve this last issue to complete the GTIN method for numerically solving the $GS{P}_3^{\left(0\right)}$ equations. Higher order base functions are derived that can provide additional parabolically weighted surface moments in order to meet all the required interface and boundary conditions. However, we discovered the fundamental problem that the presence of second order tangential derivatives in the interface/boundary condition means the condition itself being a differential equation with two degrees of freedom such that the solution is not unique but infinitely many. This issue is successfully resolved so that we can calculate the physical solution very easily despite all the possible mathematical solutions. Numerical results confirm the theoretical analysis and expectation, and demonstrate the better accuracy of $GS{P}_3^{\left(0\right)}$ versus SP₃. The method is potentially applicable to the generic geometry of arbitrary polygon. An Errata is included for correction to typos and minor errors in previous parts of this serial.