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.