Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions

We present a robust second-order accurate method for discretizing the multi-dimensional Heaviside and the Dirac delta functions on irregular domains. The method is robust in the following ways: (1) integrations of source terms on a co-dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain’s boundary; (2) the method depends only on the function value of a level function, not on its derivatives. We present the discretizations in tabulated form to make their implementations straightforward. We present numerical results in two and three spatial dimensions to demonstrate the second-order accuracy in the L1L1-norm in the case of the solution of PDEs with singular source terms. In the case of evaluating the contribution of singular source terms on interfaces, the method is also second-order accurate in the L∞L∞-norm.