A Novel High-Resolution Compact Off-Step Method via Trigonometric Spline Approximation for Second- and Fourth-Order Nonlinear Parabolic PDEs

We investigate a novel implicit two-level numerical technique of second-order accuracy in the temporal coordinate and third-order in the spatial coordinate for quasi-linear equations of parabolic type in one dimension. For discretization, we shall employ a quasi-variable mesh. The main importance of our work is that we shall obtain the method via the consistency condition of spline-in-compression estimates. The technique is indicated to be unconditionally stable, and computational results are presented to validate it. The strength of our method is quick convergence to the analytical solutions, less computational stress due to a compact cell, and simplicity and ease of the method. Further, many widespread and pertinent physical problems of fourth-order and coupled equations are computed. Their error computation with different parameters has been obtained to exhibit excellent agreement with the theoretical order presented.