Generalized Runge–Kutta method for two- and three-dimensional space–time diffusion equations with a variable time step

Solving the multigroup integro-differential equation of the neutron diffusion kinetics in 3D-Cartesian geometry

“…A homogeneous reactor (referenced in Nahla et al, 2012a) and a heterogeneous one (3D-TWIGL, referenced in Aboanber and Hamada, 2008).…”

“…Table 3 shows the results of the K eff calculation for the 3D-TWIGL reactor using the FDMs algorithm for 8 (20 Â 20 Â 20), 4 (40 Â 40 Â 40), and 2 (80 Â 80 Â 80 cells) cm-mesh size in all directions along with the results of LAPe, LAPc, TGRK (Aboanber and Hamada, 2008), and FESTRAN (Jagannathan, 1985, referenced in Aboanber andHamada, 2008). Note that the results of FDMs are approaching the results of LAPc and FESTRAN.…”

“…The 3rd application is a ramp change of the thermal removal cross section (material 1) in the TWIGL reactor as DR R2 ðtÞ ¼ À 0:0045 0:2 t for 0.2 s, it remains constant thereafter (DR R2 = À0.0045) which is described in Aboanber and Hamada (2008). Table 7 shows the BP 2 thermal flux (normalized to the value at t = 0) results for the 3rd application at six locations (p1 (80,80,16), p2(120,40,16), p3(80,80,80), p4(120,40,144), p5(80,80,144), p6 (120,40,80)) along with the results of 3DKIN, FESTRAN (a finite-element-synthesis method), and GRK-4A (a generalized Runge-Kutta method of order 4 implemented in TGRK) reported Table 4 Central flux calculation using 20 cm mesh size for an instantaneous R R2 change in the homogeneous reactor.…”

A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

“…There have been many recent studies on the numerical solution of the time dependent diffusion equation and they use new techniques such as heatlets [1], generalized Runge-Kutta method [2], a finite difference method with the stability improved [3], and a meshless solution [4]. Also, there are many studies using the finite element method based on the Galerkin's algorithm.…”

Application of Backward Differentiation Formula to Spatial Reactor Kinetics Calculation With Adaptive Time Step Control

“…However it is only second-order accurate. On the other hand, multistep methods, such as the Runge-Kutta methods, render higher accuracy as well as a provision for adaptive step size control 3 . They are not, however, practical in realistic transient calculations involving thermal-hydraulic feedback because they require multiple evaluations of solution parameters within a time step.…”