Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

Analysis of Spatial Discretization Error Estimators Implemented in ARES Transport Code for SN Equations

The discrete ordinates method (SN) is one of the mainstream methods for neutral particle transport calculations. Assessing the quality of the numerical solution and controlling the discrete error are essential parts of large-scale high-fidelity simulations of nuclear systems. Three error estimators, a two-mesh estimator, a residual-based estimator, and a dual-weighted residual estimator, are derived and implemented in the ARES transport code to evaluate the error of zeroth-order spatial discretization for SN equations. The difference in scalar fluxes on coarse and fine meshes is adopted to indicate the error in the two-mesh method. To avoid zero residual in zeroth-order discretization, angular fluxes within one cell are reconstructed by Legendre polynomials. The error is estimated by inverting the discrete transport operator using the estimated directional residual as an anisotropic source. The inner product of the forward directional residual and the adjoint angular flux is employed to quantify the error in quantities of interest which can be denoted by a linear functional of forward angular flux. Method of Manufactured Solutions (MMS) is adopted to generate analytical solutions for SN equation with scattering and the determined true error is used to evaluate the effectivity of these estimators. Promising results are obtained in the numerical results for both homogeneous and heterogeneous cases. The larger error region is well captured and the average effectivity index for the local error estimation is less than unity. For the series test problems, the estimated goal quantity error can be contained within an order of magnitude around the exact error.

IMPLEMENTATION OF EXPLICIT NORDSIECK METHODS WITH INHERENT QUADRATIC STABILITY

The present paper deals with the implementation in a variable-step algorithm of general linear methods in Nordsieck form with inherent quadratic stability and large stability regions constructed recently by Braś and Cardone. Various implementation issues such as rescale strategy, local error estimation, step-changing strategy and starting procedure are discussed. Some numerical experiments are reported, which show the performances of the methods and make comparisons with other existing methods.