A MODULUS-SQUARED DIRICHLET BOUNDARY CONDITION FOR TIME-DEPENDENT COMPLEX PARTIAL DIFFERENTIAL EQUATIONS AND ITS APPLICATION TO THE NONLINEAR SCHRODINGER EQUATION

作者：Caplan R M^*; Carretero Gonzalez R

来源：SIAM Journal on Scientific Computing, 2014, 36(1): A1-A19.

DOI：10.1137/130920046

• numerical boundary conditions
• complex partial differential equations
• nonlinear Schrodinger equation

An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-squared value of the solution at the boundaries and is defined as partial derivative Psi/partial derivative t vertical bar(b) approximate to i Im[1/Psi(b-1) partial derivative Psi/partial derivative t vertical bar(b-1)]Psi(b), where Psi is the complex field and the subscripts b and b - 1 refer to a boundary point and the closest interior point to the boundary, respectively. Application of the MSD boundary condition to simulations of the nonlinear Schrodinger equation is shown, and numerical simulations are performed to demonstrate its usefulness and advantages over other simple boundary conditions.