Numerical Preservation of Symmetry Properties of Continuum Problems

Abstract

We investigate the problem of perfectly preserving a symmetry associated naturally with one coordinate system when calculated in a different coordinate system. This allows a much wider range of problems that may be viewed as perturbations of the given symmetry to be investigated. We study the problem of preserving cylindrical symmetry in two-dimensional Cartesian geometry and spherical symmetry in two-dimensional cylindrical geometry. We show that this can be achieved by a simple modification of the gradient operator used to compute the force in a staggered grid Lagrangian hydrodynamics algorithm. In the absence of the supposed symmetry we show that the new operator produces almost no change in the results because it is always close to the original gradient operator. Our technique thus results in a subtle manipulation of the spatial truncation error in favor of the assumed symmetry but only to the extent that it is naturally present in the physical situation. This not only extends the range of previous algorithms and the use of new ones for these studies, but for spherical or cylindrical calculations it reduces the sensitivity of the results to grid setup with equal angular zoning that has heretofore been necessary with these problems.