Noniterative implicit time stepping schemes with adaptive temporal truncation error control are developed for the solution of the pressure form of Richards equation. First- and second-order linearizations of an adaptive backward Euler/Thomas-Gladwell formulation are introduced and are shown to constrain the temporal truncation errors near a user-prescribed tolerance and maintain adequate mass balance. Numerical experiments demonstrate that accurate noniterative linearizations achieve cost-effective solutions of problems where soils are described by highly nonlinear and nonsmooth constitutive functions. For these problems many conventional iterative solvers fail to converge. The noniterative formulations are considerably more efficient than analogous time stepping schemes with iterative solvers. The second-order noniterative scheme is found to be more efficient than the first-order noniterative scheme. The proposed adaptive noniterative algorithms can be easily incorporated into existing backward Euler software, which is widely used for Richards equation and other nonlinear PDEs.