A Lagrangian moving grid scheme for one-dimensional evolutionary partial differential equations
Joke G. Blom, Jesús María Sanz-Serna, Jan G. Verwer · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1987
A Lagrangian moving grid finite difference method for one-space dimensional, evolutionary partial differential equations which exhibit sharp transitions in space and time is developed.The method is based on a Crank-Nicolson type difference scheme derived via a co-ordinate transformation governed by equidistribution of the second space derivative.Each time step of our method involves two stages.First, a static grid numerical integration is carried out, immediately followed by a de Boor type redistribution of nodes at the forward time level.This stage serves only to compute the transformation.Second, a moving grid numerical integration is carried out with the Crank-Nicolson scheme.Numerical experiments show that the method automatically concentrates the grid in regions of high spatial activity and is also able to step in time with stepsizes larger than those needed by static methods, that is, methods which for intervals of time work on a fixed, nonuniform grid.As a result the method achieves high accuracy with few grid points in space and time.