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In this work we present a solution of the one-dimensional spherical symmetrictime-dependent neutron transport equation (written for a moving system inlagrangian coordinates) by using the characteristic method. One of theobjectives is to overcome the negative flux problem that arises when the systemis very opaque and the angular neutron flux can become negative when it isextrapolated in spatial meshes --- as, for example, in diamond scheme adoptedin many codes. Although there are recipes to overcome this problem, it cancompletely degrade the numerical solution if repeated many times.  The solution presented here can be easily coupled to radiation-hydrodynamicsequations, but it is necessary an additional term to maintain neutronconservation in a moving system in lagrangian coordinates. Energy multigroupmethod and a former SN method to deal with the angular variable are used, withthe assumption of isotropic scattering and the transport cross sectionsapproximation. An artifice is employed for emulating the neutron upscatteringwhen neutron energy is lower than the temperature of the medium. Theconsistency of the numerical solution is checked by making at each time-stepthe balance of neutrons in the system.  Two examples of applications are shown using aneutronic-radiation-hydrodynamic code to which the solution here presented wasincorporated: one consists of a heterogeneous pellet of DT (deuterium-tritium)tamped by an highly-enriched uranium or plutonium (a symbiotic fusion-fissionsystem); and the other is a very complex and also a symbioticfission-fusion-fission system composed by layers of the thermonuclear fuelsLiDT, LiD and a highly-supercritical fission fuel. The last is considered anextreme case for testing the time-dependent neutron transport solutionpresented here.

A Numerical Solution of the Time-Dependent Neutron Transport Equation  Using the Characteristic Method. Applications to ICF and to Hybrid  Fission-Fusion Systems