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The time‐domain Lippmann–Schwinger equation and convolution quadrature
Author(s) -
Lechleiter Armin,
Monk Peter
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21921
Subject(s) - mathematics , quadrature (astronomy) , mathematical analysis , collocation method , nyström method , domain (mathematical analysis) , time domain , convolution (computer science) , partial differential equation , integral equation , trigonometry , differential equation , ordinary differential equation , physics , computer science , machine learning , artificial neural network , optics , computer vision
We consider time‐domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solve a time‐domain volume Lippmann–Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space, we can compute an approximate solution. We prove that the time‐domain Lippmann–Schwinger equation has a unique solution and prove conditional convergence and error estimates for the fully discrete solution for globally smooth sound speeds. Preliminary numerical results show that the method behaves well even for discontinuous sound speeds. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 517–540, 2015