Stormer-Numerov approximations of high accuracy were developed for solutions of nonlinear boundary value problems and nonlinear elliptic partial differential equations. The approximations can be easily adopted also for parabolic partial differential equations in one and more space dimensions and feature fourth-order accuracy. For boundary value problems only three nodes are necessary to obtain the desired fourth order accuracy. The finite difference formula for parabolic partial differential equations can be readily generalized to a nonequidistant mesh so that automatic regridding in space may be used. The Stomer-Numerov approximations are important for solution of problems where storage limitations and computer time expenditure preclude standard second order methods. Because of the fourth order approximations a low number of mesh points can be used for a majority of chemical engineering problems. The application of Stormer-Numerov approximations is illustrated on a number of examples.