Stockpile stewardship refers to the United States program of reliability testing and maintenance of its nuclear weapons without the use of nuclear testing.
Because no new nuclear weapons have been developed by the United States since 1992, even its youngest weapons are at least 19 years old. Aging weapons can fail or act unpredictably in a number of ways: the high explosives that condense their fissile material can chemically degrade, their electronic components can suffer from decay, their radioactive plutonium/uranium cores are potentially unreliable, and the isotopes used by thermonuclear weapons may be chemically unstable as well.
Since the United States has also not tested nuclear weapons since 1992, this leaves the task of its stockpile maintenance resting on the use of simulations (using non-nuclear explosives tests and supercomputers, among other methods) and applications of scientific knowledge about physics and chemistry to the specific problems of weapons aging (the latter method is what is meant when various agencies refer to their work as "science-based"). It also involves the manufacture of additional plutonium "pits" to replace ones of unknown quality, and finding other methods to increase the lifespan of existing warheads and maintain a confident nuclear deterrent.
Most work for stockpile stewardship is undertaken at United States Department of Energy National Laboratories, mostly at Los Alamos National Laboratory, Sandia National Laboratories, Lawrence Livermore National Laboratory, the Nevada Test Site, and Department of Energy productions facilities, which employ around 27,500 personnel and cost billions of dollars per year to operate.
Read more about Stockpile Stewardship: Stockpile Stewardship and Management Program