| Abstract: |
| This paper deals with the initial-value problem for the radially symmetric nonlinear Schrodinger equation with cubic non linearity in d = 2 and 3 space dimensions. A very simple, robust and efficient moving mesh method is proposed for numerically solving the radially symmetric nonlinear Schrodinger equation. It is employed in two schemes. The first numerical simulation aims to reproduce the stable self-similar blow-up solution for d=3. The computed data is used to compare it with the exact blow-up solution. The two solutions overlap when the amplitude of the solution is less than 10 to the 5th power. Next, a typical initial function is used to simulate the blow-up solution for d=3 and is compared with the corresponding exact blow-up solution. The graphs of the two solutions almost overlap when the amplitude of the solution reaches 10 to the 60th power and the adjacent mesh points near 0 are as small as 10 to the -61th power. The two solution curves and their derivatives are smooth in the whole domain and show slow oscillations in both r and t directions. However, when d = 2, the numerical solution becomes unstable due to high oscillations. A comparison with the corresponding asymptotic solution reveals that the amplitude of the two solutions almost overlap. Furthermore, both mass and energy are well conserved for d = 2 and 3. |
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