The equations of the high-order linear-nodal numerical scheme are cast in an augmented weighted-difference form for three-dimensional Cartesian nodes. The coupling exhibited by these equations indicates that this new algorithm is simpler and, hence, faster than previous nodal schemes of this degree of accuracy. A well-logging problem and a fast reactor problem are examined. The new scheme developed is compared with the classical linear-linear nodal scheme and the diamond-difference scheme. For the well-logging problem, it is found that the new scheme is both faster and simpler than the classical linear-linear nodal scheme while sacrificing little in accuracy. Even though the new scheme is more accurate than the diamond-difference scheme for the reactor problem, the results indicate that state-of-the-art acceleration methods are needed for nodal schemes.