In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.
Integrable nonlocal derivative nonlinear Schrödinger equations
Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation.
- Award ID(s):
- 2005343
- Publication Date:
- NSF-PAR ID:
- 10335887
- Journal Name:
- Inverse Problems
- Volume:
- 38
- Issue:
- 6
- Page Range or eLocation-ID:
- 065003
- ISSN:
- 0266-5611
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Publisher's Version of Record
- https://doi.org/10.1088/1361-6420/ac5f75
