Towards Accurate Artificial Boundary Conditions for Nonlinear PDEs Through Examples

The aim of this paper is to give a comprehensive review of current developments related to the derivation of artificial boundary conditions for nonlinear partial differential equations. The essential tools to build such boundary conditions are based on pseudodifferential and paradifferential calculus. We present various derivations and compare them. Some numerical results illustrate their respective accuracy and analyze the potential of each technique.

[1]  Jeremie Szeftel.  Calcul pseudodifférentiel et paradifférentiel pour l'étude de conditions aux limites absorbantes et de propriétés qualitatives d'équations aux dérivées partielles non linéaires , 2004 .

[2]  J. Bony,et al.  Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , 1980 .

[3]  Thomas Hagstrom,et al.  On the accurate long-time solution of the wave equation in exterior domains: asymptotic expansions and corrected boundary conditions , 1994 .

[4]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[5]  Jérémie Szeftel,et al.  A nonlinear approach to absorbing boundary conditions for the semilinear wave equation , 2006, Math. Comput..

[6]  Jeremie Szeftel,et al.  Absorbing boundary conditions for nonlinear scalar partial differential equations , 2006 .

[7]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[8]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[9]  T. Hagstrom.  Radiation boundary conditions for the numerical simulation of waves , 1999, Acta Numerica.

[10]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[11]  Houde Han,et al.  Absorbing boundary conditions for nonlinear Schrödinger equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  B. Engquist,et al.  Long-time behaviour of absorbing boundary conditions , 1990 .

[13]  Andrew J. Majda,et al.  Reflection of singularities at the boundary , 1975 .

[14]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[15]  A. Durán,et al.  The numerical integration of relative equilibrium solutions. Geometric theory , 1998 .

[16]  M. Czubak,et al.  PSEUDODIFFERENTIAL OPERATORS , 2020, Introduction to Partial Differential Equations.

[17]  Christophe Besse,et al.  Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation , 2003 .

[18]  Chunxiong Zheng,et al.  Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations , 2006, J. Comput. Phys..

[19]  Marcus J. Grote,et al.  On nonreflecting boundary conditions , 1995 .

[20]  G. Mur.  Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations , 1981, IEEE Transactions on Electromagnetic Compatibility.

[21]  A. Durán,et al.  The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation , 2000 .

[22]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[23]  Christophe Besse,et al.  Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations , 2006, SIAM J. Numer. Anal..