A new 2-level compact off-step implicit method in exponential form for the solution of fourth order nonlinear parabolic equations

In this work, we describe a new two-level compact implicit strategy in exponential form for the numerical solution of a one-dimensional quasi linear parabolic equation. The method is based on off-step variable (or constant) mesh discretization and of order of accuracy two in time direction while three (or four) in space direction. At each level of time, the method uses just three adjacent grid points, and the boundary restrictions are met precisely, with no additional requirements at the edges. The approach is immediately applicable to singular equations and extends its applications to the coupled Burgers' equation and nonlinear fourth order PDEs such as Fisher-Kolmogorov equations. Stability of the method is discussed and shown unconditionally stable when applied to the fourth order model linear problem. When the resulting findings are compared to those of some other previously known methods, the current method is clearly superior.