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Discretizing delta functions via finite differences and gradient normalization

机译：通过有限差分和梯度归一化离散化增量函数

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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 {mapping} R. These methods were shown to be consistent (meaning that they converge to the true solution as the mesh size h → 0) in the codimension one setting. In this paper, we concentrate on n ≤ 3, but generalize our methods to codimensions other than one - now the level set function is generally a vector valued mapping over(u, →): R~n {mapping} R~m, 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 (h~2) convergence for our new gradient-normalized FDM2 algorithm. In the full codimension setting (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.
机译：在[J.D.塔，J。Comput，级别集上支持的两种离散量增量函数的方法。物理220（2007）915-931]作者提出了两种密切相关的有限差分方法（在此称为FDM1和FDM2），用于离散化支持由一维流形的零级集定义的一维流形上支持的增量函数： Rn {mapping}R。这些方法被证明是一致的（这意味着它们在网格维数h→0时收敛于真实解）在余维一设置中。在本文中，我们集中在n≤3上，但是将我们的方法推广到除一个之外的余维上-现在，水平集函数通常是映射到（u，→）的向量值：R〜n {mapping} R〜m，1 ≤m≤n≤3。当将这些问题应用于基于近似德尔塔函数的简单乘积的看似合理的算法时，通常并不一致。因此，我们改用楔积积形式来概括我们的FDM算法，这种方法可以得出准确的，通常是一致的近似值。为了总体上确保一致性，我们建议在FDM算法之前应用新的梯度归一化过程。这些组合算法在所有合理情况下似乎都是一致的，数值实验表明我们的新梯度归一化FDM2算法具有O（h〜2）收敛性。在完整的余维设置（m = n）中，当使用更多标准的近似增量函数时，我们的梯度归一化处理还可以提高精度。这种组合也可以得出近似一致的近似值。

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