Abstract <p> Compact and monotone difference schemes of the fourth order of accuracy preserving theconservativeness (divergence) property for the one- and two-dimensional quasilinear stationaryreaction-diffusion equations are constructed and investigated. A priori estimates of the differencesolution in the nonlinear case for the one-dimensional quasilinear equation are obtained based onthe established two-sided estimates of the grid solution. For the linearization of the nonlineardifference scheme, an iterative method of the Newton–Seidel type preserving conservativeness andmonotonicity is used. The main idea of the proposed difference schemes is based on the possibilityof parallelizing the computational process. The emerging problems of finding additional boundaryconditions at boundary nodes in both one- and two-dimensional cases are solved using the Newtoninterpolation polynomial of the fourth order of accuracy. The presented results of thecomputational experiments illustrate the increased order of the proposed algorithms. Thepossibility of generalizing this method to nonstationary quasilinear equations is also indicated.</p>