The solution of large-scale, nonlinear PDE-based simulations typically depends on the performance of sparse linear solvers, which may be invoked at each nonlinear iteration. We present a framework for using multi-method solvers in such simulations to potentially improve the execution time and reliability of linear system solution. We consider composite solvers, which provide reliable linear solution by using a sequence of preconditioned iterative methods on a given system until convergence is achieved. We also consider adaptive solvers, where the solution method is selected dynamically to match the attributes of linear systems as they change during the course of the nonlinear iterations. We demonstrate how such multi-method composite and adaptive solvers can be developed using advanced software architectures such as PETSc, and we report on their performance in a computational fluid dynamics application.