A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schrödinger Equation

Abstract

Abstract. 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 1. Introduction. When utilizing numerical methods to approximate the solutions to time-dependent partial differential equations (PDEs), proper handling of boundary conditions can be quite challenging. Sometimes, an otherwise stable numerical scheme will become unstable depending on how the boundary conditions are computed [42]. In addition, high-order schemes can degrade in accuracy to lower order when using boundary conditions which are not compatible with the high-order accuracy [26]. Proper handling of boundary conditions in higher-order schemes, especially in high-order compact schemes, can be even more of a challenge [19,20].Often, researchers will forgo a complicated boundary condition implementation and instead use tried-and-true boundary condition techniques which are very simple yet provide acceptable results. One of the most common is the use of Dirichlet boundary conditions when simulating solutions which decay towards zero at infinity, and where most of the dynamics (or "action") is expected to remain in the central regions of the computational grid. Another simple method in such cases is to use periodic boundary conditions.Infinite-domain problems involving PDEs whose function values are complex cannot, in general, make use of numerical Dirichlet or periodic boundary conditions because of the oscillation of the real and imaginary parts of the function due to the intrinsic frequency of the system. (In cases when both the real and imaginary parts of the solution converge to a constant at infinity, such boundary conditions can be

Numerical stability of explicit Runge–Kutta finite-difference schemes for the nonlinear Schrödinger equation

“…where C is a constant. We have recently formulated a method for such a boundary condition (which is almost as easy to implement as Dirichlet) called the modulus-squared Dirichlet boundary condition [5]. The MSD boundary condition is given in terms of the temporal derivative of the NLSE as…”

A two-step high-order compact scheme for the Laplacian operator and its implementation in an explicit method for integrating the nonlinear Schrödinger equation

“…There are some boundary condition techniques that can be seen as compact, and hence fit very well into the framework of HOC schemes. These conditions are a Dirichlet (Ψ = const) and modulus-squared Dirichlet (MSD) (|Ψ| 2 = const) [35] boundary conditions.…”

“…In such a case, one cannot use standard Dirichlet boundary conditions since the constant boundary is in the modulus-squared of the wavefunction, not at a single real and imaginary value. To solve this problem, we recently have developed a new modulus-squared Dirichlet boundary condition that accurately simulates a constant density at the boundaries [35]. This boundary condition is described as…”

“…The standard form of the MSD boundary condition requires to first compute all interior points and then compute the boundaries, making the general form of the MSD not compatible with implicit schemes (see Ref. [35] for details on possible implicit scheme implementation strategies of the MSD). The MSD boundary condition can be used for any time-dependent complex PDE and in multiple dimensions.…”

Scattering and leapfrogging of vortex rings in a superfluid

“…Since the stability forces the timestep to be proportional to h 2 , the overall error of the scheme is O(h 4 ). Due to the constant density background of the problem, we utilize a recently developed modulussquared Dirichlet boundary condition [48]. The simulations are computed using the NLSEmagic code package [49] [78] which contains algorithms written in C and CUDA, and are primarily run on NVIDIA GeForce GTX 580 and GeForce GT 650M GPU cards.…”