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 variational level set approach to multiphase motion
Journal of Computational Physics
A PDE-based fast local level set method
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
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
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
A convergence rate theorem for finite difference approximations to delta functions
Journal of Computational Physics
High Order Numerical Quadratures to One Dimensional Delta Function Integrals
SIAM Journal on Scientific Computing
Journal of Computational Physics
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
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In this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case. The methods comprise approximating the mesh cell restrictions of the delta function integral. In each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral, and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in [X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys. 226 (2007) 1952-1967]. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally. Numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods.