Numerical Approximations of Stochastic Maxwell Equations

Chuchu Chen is an associate professor at the Chinese Academy of Sciences. She obtained her Ph.D. in 2015 at the Chinese Academy of Sciences. Her research interest is in the numerical analysis of stochastic partial differential equations, especially in the structure-preserving algorithms for stochastic Hamiltonian PDEs including the stochastic Maxwell equations and the stochastic Schrödinger equation, the analysis of the long-time dynamical behaviors including the ergodicity and intermittency of stochastic numerical methods, the influence of numerical discretizations on the statistical properties like the hitting probability and density function of stochastic PDEs.Jialin Hong is a professor at the Chinese Academy of Sciences. He obtained his Ph.D. in 1994 at Jilin University. He works in various directions including structure-preserving algorithms for dynamical systems involving symplectic and multi-symplectic methods for Hamiltonian ODEs andPDEs, Lie group methods and applications, numerical dynamics including chaos, bifurcations for discrete systems, numerical methods for stochastic ordinary differential systems, stochastic partial differential equations, and backward stochastic differential equations, almost periodic dynamical systems, and ergodic theory.Lihai Ji is an associate professor at the Institute of Applied Physics and Computational Mathematics. He obtained his Ph.D. in 2013 at the Chinese Academy of Sciences. He works in stochastic partial differential equations and their numerical algorithms. He has been investigating the construction and analysis of various energy-preserving algorithms, positive-preserving algorithms, stochastic symplectic and multi-symplectic algorithms for the stochastic Lotka–Volterra model and stochastic Hamiltonian PDEs including the stochastic Maxwell equations, the stochastic Schrödinger equation, and the coupled stochastic Schrödinger equation.

• Introduction.- Solution Theory of Stochastic Maxwell Equations.- Intrinsic Properties of Stochastic Maxwell Equations.- Structure-Preserving Algorithms for Stochastic Maxwell Equations.- Convergence Analysis of Structure-Preserving Algorithms.- Implementation of Numerical Experiments.- Appendix A: Basic Identities and Inequalities.- Appendix B: Semigroup, Sobolev Space, and Differential Calculus.- Appendix C: Estimates Related to Maxwell Operators.- Appendix D: Some Results of Stochastic Partial Differential Equations.- References.

“This interesting monograph brings a self-contained account of the numerical analysis of the stochastic Maxwell equations. More precisely, it includes the construction and analysis of structure-preserving algorithms with an emphasis on the preservation of geometric structures, physical properties, and asymptotic behavior of the original equations.” (Jaromír Antoch, zbMATH 1542.65001, 2024)