Finite Differences Method and Adaptive Grids in the Method of Lines for Partial Differential Equation

In this paper a time-dependent moving-grid method is described to numerically solve time-dependent partial differential equations (PDEs) in one space dimensions involving fine scale structures such as steep moving fronts, emerging steep layers, pulses and shocks. Smoothing in the spatial direction is employed to control grid clustering and expansion. Additional smoothing in the temporal direction ensures a smooth progression of the grid points in time by preventing the points from responding too quickly to current values of the weight functions. In particular, we focus attention on finite differences scheme and adaptative grids using Method Of Line (MOL) toolbox within MATLAB. The numerical simulation includes various spatial approximation schemes based on finite differences and slope limiters. Several finite difference schemes, are compared. The performance of the algorithm is demonstrated with illustrative example including, a model of flame propagation, a methanisation in a reactor problem, and a classical Korteweg-de-Vries equation.