Solution of the one-dimensional transport equation by the vector Green function method: Error bounds and simulation

•We solve numerically the anisotropic transport equation using techniques of integral operators.•We present bound error estimates for the GFD method.•We present numerical results for one dimensional transport equation near criticality.•We apply the Green function decomposition method to solve numerically the radiative transport equation in a slab.

In this work we solve the general anisotropic transport equation for an arbitrary source with semi-reflexive boundary conditions. First we present a complete existence theory for this problem in the space of continuous functions and in the space of α-Hölder continuous functions. As a result of our analysis we construct integral operators which we discretize in a finite dimensional functional space, yielding a new robust numerical method for the transport equation, which we call Green’s function decomposition method (GFD). As well, we demonstrate a convergence theorem providing error bounds for the reported method. Finally we provide numerical results and applications.