HIDM, a general purpose algorithm for nonlinear problems described by partial differential equations A new numerical scheme HIDM (higher-order implicit difference method) is developed to solve comprehensively algebraic equations, ordinary differential equations, partial differential equations and coupled systems of these equations under high accuracy and high numerical stability. High accuracy of the scheme is realized by higher-order discretization method and high numerical stability of the scheme is obtained by implicit difference method and tournament multidivided shooting scheme.par A general purpose computer code HIDM2D is constructed based on the numerical scheme for variety of the time evolution of boundary value problems (1D spatial variable + time). High performance of the code is demonstrated by solving nonlinear dissipative equations, nonlinear dispersive equations, and nonlinear dynamics of unstable systems.