4. The Method of Nearby Problems#
The neutron transport equation is solved numerically through both deterministic (e.g. discrete ordinates) and probabilistic (e.g. Monte Carlo) methods [LM93]. Two types of verification are used to ensure that these simulations are approximating the solution correctly. The first type is code verification [Roy05]. This is a process to ensure that the numerical discretization used by computer codes is functioning properly. One such technique is the method of manufactured solutions (MMS), which starts with an analytical solution and works backwards to solve for the initial and boundary conditions of the problem. The initial and boundary conditions are used to create a numerical approximation to the known “manufactured” solution. Several numerical approximations with different mesh sizes are used to estimate the discretization error and verify the correct rate of convergence. To prevent requiring significant amounts of time to formulate these analytical solutions and their associated initial and boundary conditions, simple, and often nonphysical, solutions are used. This technique has been implemented numerous times with neutron transport solvers [Wan19].
The second type of verification is solution verification. This is a process to ensure that the discretization error associated with a given numerical solution is sufficiently close to the true solution of the governing equation [Roy05]. As opposed to the simple solutions from the method of manufactured solutions, solution verification is often used with complex problems and is implemented when providing engineering analysis. In most instances, the true solution to the governing equation is neither known nor readily available. For these cases, techniques such as extrapolation-based error estimation are used. One extrapolation-based error estimation employs Richardson extrapolation to approximate a higher-order numerical solution to estimate the correct discretization error [Roy05]. This method is effective at determining the global error for a problem, but requires at least two (high- and low-fidelity) grids. In terms of estimating spatial error, which this chapter focuses on, using multiple spatial grids can become difficult in instances with unstructured meshes. The ability to develop separate high- and low-fidelity unstructured grids might not be obtainable and can require significant computational and analyst time [Mav97].
Rather than requiring multiple spatial grids, the method of nearby problems [RRH07] uses single-grid error estimation for solution verification. After solving for a numerical solution, an analytical function can be constructed by interpolating with a known polynomial function. The residual between the analytical solution and the numerical solution is calculated and added to the original problem as an additional source term. Utilizing the additional source term on the same spatial grid calculates the nearby solution. The nearby solution can be compared to the analytical solution and is used to estimate the discretization error. Additional measures must be taken to calculate the spatial discretization error with the method of nearby problems for \(k\)-eigenvalue problems. This technique has been proven to work with various PDE problems [RRH07] and expands upon previous work [WMHS23] as it applies to the neutron transport equation.
This chapter is organized as follows. The fixed-source formulation explains the method of nearby problems as it pertains to fixed-source problems and how it is directly applied to the multigroup neutron transport equation. The k-eigenvalue formulation discusses the application of the method to \(k\)-eigenvalue problems. The Hermite spline curve fit and residual formulation then detail the construction of the curve-fit flux and the discretization of the residual introduced in the fixed-source formulation.