The Rothe-Newton approach to simulate the variable coefficient convection-diffusion equations

The current article presents a novel hybrid approach based on the Rothe time-marching algorithm and a spectral matrix collocation approach using the well-known Newton bases to deal with the spatial variable. Utilizing the Rothe approach converts the underlying convection-diffusion into initial-boundary value problems and then the Newton collocation method solves the continuous discretized time equation in each time step. The error analysis of the newly employed basis functions is established. Three numerical simulations are developed to show the accuracy and utility of the proposed hybrid strategy.Applying the current study to other linear and nonlinear PDEs and high-order PDEs can be performed straightforwardly.