An improved variational nodal method for the solution of the three-dimensional steady-state multi-group neutron transport equation

Highlights: • The new method presents asymptotic convergence trends as the standard variational nodal method while increasing angular and spatial orders. Same level of accuracies can be achieved upon asymptotic convergence. However, the new method yields more rapid convergence while angular orders are increased. • For the TAKEDA benchmark, with P₁ angular approximations, the new method reduces the error of eigenvalues by an order of magnitude and the error of fluxes by 5%. • For the TAKEDA benchmark, with P₉ the gains by using the new method are 56% in the memory storage and 81% in the response matrix formation time. • For the 3D PWR problem, the GPM method reduces the total number of outer iterations from 104 to 10, resulting in a speed-up ratio of 20.6 when combined with the legacy PM acceleration method. - Abstract: An improved variational nodal method is presented for the solution of the three-dimensional (3D) steady-state multi-group neutron transport equation. The variational functional is constructed that reproduces the even parity neutron transport equation with isotropic scattering. 3D orthogonal polynomials are used to approximate the spatial flux distribution within the nodes and across the nodal interfaces. The angular discretization utilizes a 3D even-parity integral method within the nodes, and standard spherical harmonics (P_N) on the interfaces. The generalized partitioned matrix (GPM) acceleration is derived and performed to speed up outer iterations of the transport formulation. Examined against three sets of TAKEDA benchmark cases, the integral method exhibits superior accuracy and efficiency than the standard VNM approach. In addition, the GPM method presents remarkable acceleration to outer iterations when tested on the 3D PWR problem. The joint employment of the GPM method and the PM method yields a gain of over 20 in the response matrix solution time.