Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity
In this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserves both mass and energy through provable conservation propert...
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2025-06-01
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| author | He Yang |
| author_facet | He Yang |
| author_sort | He Yang |
| collection | DOAJ |
| description | In this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserves both mass and energy through provable conservation properties. Under suitable assumptions on the exact solution, we establish upper and lower bounds for the numerical solution in the infinity norm, and further prove that the errors are fourth-order accurate in space and second-order in time in both the 2-norm and infinity norm. A detailed description of the nonlinear system solver at each time step is provided. We validate the proposed method through numerical experiments that demonstrate its efficiency, including fourth-order convergence (when sufficiently small time steps are used) and machine-level accuracy in the relative errors of mass and energy. |
| format | Article |
| id | doaj-art-dde7ab23bde342f6a564747349366dc2 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-dde7ab23bde342f6a564747349366dc22025-08-20T03:27:15ZengMDPI AGMathematics2227-73902025-06-011312197810.3390/math13121978Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic NonlinearityHe Yang0Department of Mathematics, Augusta University, 1120 15th Street, Augusta, GA 30912, USAIn this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserves both mass and energy through provable conservation properties. Under suitable assumptions on the exact solution, we establish upper and lower bounds for the numerical solution in the infinity norm, and further prove that the errors are fourth-order accurate in space and second-order in time in both the 2-norm and infinity norm. A detailed description of the nonlinear system solver at each time step is provided. We validate the proposed method through numerical experiments that demonstrate its efficiency, including fourth-order convergence (when sufficiently small time steps are used) and machine-level accuracy in the relative errors of mass and energy.https://www.mdpi.com/2227-7390/13/12/1978compact finite difference methodSchrödinger equation with anti-cubic nonlinearityerror estimatesconservative numerical methods |
| spellingShingle | He Yang Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity Mathematics compact finite difference method Schrödinger equation with anti-cubic nonlinearity error estimates conservative numerical methods |
| title | Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity |
| title_full | Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity |
| title_fullStr | Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity |
| title_full_unstemmed | Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity |
| title_short | Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity |
| title_sort | fourth order compact finite difference method for the schrodinger equation with anti cubic nonlinearity |
| topic | compact finite difference method Schrödinger equation with anti-cubic nonlinearity error estimates conservative numerical methods |
| url | https://www.mdpi.com/2227-7390/13/12/1978 |
| work_keys_str_mv | AT heyang fourthordercompactfinitedifferencemethodfortheschrodingerequationwithanticubicnonlinearity |