Compact difference schemes for one-dimensional quasilinear parabolic equations

Compact finite difference schemes of approximation orders 4 + 1 and 4 + 2, constructed on minimal stencils, are presented and investigated for the one-dimensional non-stationary quasilinear heat equation, and do not require an iterative process for their implementation. The computational efficiency is achieved by parallelizing the Thomas algorithm over even and odd grid nodes. The monotonicity conditions are obtained and two-sided estimates of the difference solution and a priori estimates in the uniform norm are proved. Computational experiments are also presented to illustrate the effectiveness of the proposed methods, as well as their convergence with the corresponding order.