A front-tracking method for viscous, incompressible, multi-fluid flows
Journal of Computational Physics
A level set approach for computing solutions to incompressible two-phase flow
Journal of Computational Physics
A fast level set method for propagating interfaces
Journal of Computational Physics
A front-tracking method for dendritic solidification
Journal of Computational Physics
A variational level set approach to multiphase motion
Journal of Computational Physics
A PDE-based fast local level set method
Journal of Computational Physics
A front-tracking method for the computations of multiphase flow
Journal of Computational Physics
Motion of curves in three spatial dimensions using a level set approach
Journal of Computational Physics
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
Construction of Shapes Arising from the Minkowski Problem Using a Level Set Approach
Journal of Scientific Computing
Regularization Techniques for Numerical Approximation of PDEs with Singularities
Journal of Scientific Computing
Local level set method in high dimension and codimension
Journal of Computational Physics
Numerical approximations of singular source terms in differential equations
Journal of Computational Physics
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
The numerical approximation of a delta function with application to level set methods
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Two methods for discretizing a delta function supported on a level set
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Geometric integration over irregular domains with application to level-set methods
Journal of Computational Physics
High order numerical methods to a type of delta function integrals
Journal of Computational Physics
Short Note: A proof that a discrete delta function is second-order accurate
Journal of Computational Physics
Journal of Computational Physics
A convergence rate theorem for finite difference approximations to delta functions
Journal of Computational Physics
Journal of Scientific Computing
High Order Numerical Quadratures to One Dimensional Delta Function Integrals
SIAM Journal on Scientific Computing
Journal of Computational Physics
Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime
Journal of Computational Physics
High order numerical methods to two dimensional delta function integrals in level set methods
Journal of Computational Physics
A Hybrid Phase-Flow Method for Hamiltonian Systems with Discontinuous Hamiltonians
SIAM Journal on Scientific Computing
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials
Journal of Computational Physics
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In this paper we propose a class of high order numerical methods to delta function integrals appearing in level set methods in the three dimensional case by extending the idea for designing high order methods to two dimensional delta function integrals in [X. Wen, J. Comput. Phys., 228 (2009), pp. 4273-4290]. The methods comprise approximating the mesh cell restrictions of the delta function integral. In each mesh cell the three dimensional delta function integral can be rewritten as a two dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral. A key idea in designing high order methods in this paper is that the high order accuracy of the methods is not ensured in each mesh cell, and the methods are designed in a consistent way to foster error cancelations in mesh cells where high order accuracy is not ensured in the individual cell. This issue is essentially related to the construction of integral area for the two dimensional ordinary integral in each mesh cell, which is achieved by considering the intersection points between the zero level set and the edges or sides of the mesh cell. The mesh cell restrictions of the three dimensional delta function integral are then approximated by applying standard two dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals. Numerical examples are presented showing that our methods in this paper achieve or exceed the expected second to fourth order accuracy and demonstrating the stability of shifting mesh for our methods and the advantage of using our high order numerical methods.