A finite difference moving mesh method based on conservation for moving boundary problems

We propose a velocity-based moving mesh method in which we move the nodes so as to preserve local mass fractions. Consequently, the mesh evolves to be finer where the solution presents rapid changes, naturally providing higher accuracy without the need to add nodes. We use an integral approach which avoids altering the structure of the original equations when incorporating the velocity and allows the solution to be recovered algebraically. We apply our method to a range of one-dimensional moving boundary problems: the porous medium equation, Richards’ equation, and the Crank–Gupta problem. We compare our results to exact solutions where possible, or to results obtained from other methods, and find that our approach can be very accurate (1% relative error) with as few as ten or twenty nodes.