Numerical solution of the electron transport equation in the upper atmosphere

Highlights: • The electron transport equation is given, and special attention is given to the boundary conditions, which cause difficulty. • An upwind method is given in detail showing how to overcome the exponential stiffness. • A simplified problem that has an exact solution but still retains the high stiffness verifies the upwind method. -- Abstract: A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplified problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other, verifying the method.