Date of Award
Master of Science in Applied Mathematics
Department of Mathematics and Statistics
Jonah A. Reeger, PhD.
Solutions to the one-dimensional and two-dimensional nonlinear Schrodinger (NLS) equation are obtained numerically using methods based on radial basis functions (RBFs). Periodic boundary conditions are enforced with a non-periodic initial condition over varying domain sizes. The spatial structure of the solutions is represented using RBFs while several explicit and implicit iterative methods for solving ordinary differential equations (ODEs) are used in temporal discretization for the approximate solutions to the NLS equation. Splitting schemes, integration factors and hyperviscosity are used to stabilize the time-stepping schemes and are compared with one another in terms of computational efficiency and accuracy. This thesis shows that RBFs can be used to numerically solve the NLS with reasonable accuracy. Integration factors and splitting methods yield improvements in stability at the cost of computation time; both methods produce solutions of similar accuracy while splitting methods are slightly less expensive to implement than integration factors (computation times were of the same order of magnitude). The use of hyperviscosity can lead to an improvement in stability but can also lead to increased errors if the relevant parameters are not chosen carefully.
DTIC Accession Number
Ng, Justin, "Radial Basis Function Generated Finite Differences for the Nonlinear Schrodinger Equation" (2018). Theses and Dissertations. 1740.