Efficient, linearized high-order compact difference schemes for nonlinear parabolic equations I: One-dimensional problem

A one-dimensional nonlinear time-dependent conservative convection-diffusion-reaction equation is considered, some linearized high-order compact (HOC) difference schemes are developed and analyzed. Laplace-modified method and implicit-explicit high-order approximation technique are considered for the temporal discretization. While for the spatial discretization, the so-called high-order narrow stencil approximations are employed for the nonlinear second-order spatial derivative, and the HOC difference approximations are considered, respectively, for the nonlinear first-order spatial derivative and the introduced Laplace term. At each time level, the resulting algebraic system arising from the proposed schemes is linear and has the same (periodic) tridiagonal constant-coefficient matrix, even though the convection-diffusion-reaction equation itself has time-dependent variable coefficients or even strongly nonlinear coefficients. Therefore, the proposed schemes are easy to implement in only O(M)$$ \mathcal{O}(M) $$ computational complexity, where M$$ M $$ is the number of spatial unknowns per time level. The presented high-order schemes are proved to be unconditionally stable in the linear case, and optimal-order error estimates for the nonlinear equation in the discrete H-1-norm are obtained under reasonable restrictions on the temporal stepsize ratio by using the inductive arguments. Finally, ample numerical experiments are carried out to show the high accuracy and efficiency of the schemes.