A two-zone method with an enhanced accuracy for a numerical solution of the diffusion equation

A variational principle is applied to the diffusion equation to numerically obtain the fission gas release from a spherical grain. The two-zone method, originally proposed by Matthews and Wood, is modified to overcome its insufficient accuracy for a low release. The results of the variational approaches are examined by observing the gas concentration along the grain radius. At the early stage, the concentration near the grain boundary is higher than that at the inner points of the grain in the cases of the two-zone method as well as the finite element analysis with the number of the elements at as many as 10. The accuracy of the two-zone method is considerably enhanced by relocating the nodal points of the two zones. The trial functions are derived as a function of the released fraction. During the calculations, the number of degrees of freedom needs to be reduced to guarantee physically admissible concentration profiles. Numerical verifications are performed extensively. By taking a computational time comparable to the algorithm by Forsberg and Massih, the present method provides a solution with reasonable accuracy in the whole range of the released fraction.