High order finite difference approximations to the one-group neutron diffusion equation in 1D heterogeneous media part I: Theory in plane media

Verification that a numerical method performs as intended is an integral part of code development. Semi-analytical benchmarks enable one such verification modality. Unfortunately, a semi-analytical benchmark requires some degree of analytical forethought and treats only relatively idealized cases making it of limited diagnostic value. In the first part of our investigation (Part I), we establish the theory of a straightforward finite difference scheme for the 1D, monoenergetic neutron diffusion equation in plane media. We also demonstrate an analytically enhanced version that leads directly to the analytical solution. The second part of our presentation (Part II, in these proceedings) is concerned with numerical implementation and application of the finite difference solutions. There, we demonstrate how the numerical schemes themselves provide the semi-analytical benchmark. With the analytical solution known, we therefore have a test for accuracy of the proposed finite difference algorithms designed for high order.