Efficient and Stable Method to Solve Poisson–Boltzmann Equation with Steep Gradients

Abstract

This work describes a method to solve Poisson–Boltzmann Equation (PBE) with steep gradients when common methods of solving non-linear equations do not converge or work inefficiently. This is particularly the case for relatively large boundary conditions leading to rapid changes of the solution in a narrow interval. The method uses adaptive step in the region of abrupt change of the solution and adopts invertible mapping algorithm to transform the original PBE into a form with a smooth profile which ensures convergency and stability of the solution. The numerical tests demonstrate the advantages of the method compared to usual successive iteration methods, in which the solution is gradually approached by iterations from small values.