Discretizing delta functions via finite differences and gradient normalization

In [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915–931] the author presented two closely related finite difference methods (referred to here as FDM1 and FDM2) for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u:Rn↦R. These methods were shown to be consistent (meaning that they converge to the true solution as the mesh size h→0h→0) in the codimension one setting.In this paper, we concentrate on n⩽3n⩽3, but generalize our methods to codimensions other than one – now the level set function is generally a vector valued mapping u→:Rn↦Rm,1⩽m⩽n⩽3. Seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems. Motivated by this, we instead use the wedge product formalism to generalize our FDM algorithms, and this approach results in accurate, often consistent approximations. With the goal of ensuring consistency in general, we propose a new gradient normalization process that is applied before our FDM algorithms. These combined algorithms seem to be consistent in all reasonable situations, with numerical experiments indicating O(h2)O(h2) convergence for our new gradient-normalized FDM2 algorithm.In the full codimension setting (m=n)(m=n), our gradient normalization processing also improves accuracy when using more standard approximate delta functions. This combination also yields approximations that appear to be consistent.