论文标题
非线性Schrödinger方程的松弛有限差异方案的错误估计
Error estimation of the Relaxation Finite Difference Scheme for the nonlinear Schrödinger Equation
论文作者
论文摘要
在一个空间维度情况下，我们考虑了具有均匀的dirichlet边界条件的非线性schrödinger方程的初始和边界价值问题。我们通过中央有限差分法和及时通过C. Besse提出的放松方案来离散空间中的问题。 R. Acad。科学。巴黎塞尔。 i {\ bf 326}（1998），1427-1432]。我们在离散$ l_t^{\ infty}（h_x^1）$ norm中提供最佳订单错误估计值，以在时间节点和中间时间节点时的近似错误。在非线性schr {Ö} dinger方程的上下文中，这是第一次完全解决了基于松弛方案的完全离散方法的误差估计值。
We consider an initial- and boundary- value problem for the nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference method and in time by the Relaxation Scheme proposed by C. Besse [C. R. Acad. Sci. Paris Sér. I {\bf 326} (1998), 1427-1432]. We provide optimal order error estimates, in the discrete $L_t^{\infty}(H_x^1)$ norm, for the approximation error at the time nodes and at the intermediate time nodes. In the context of the nonlinear Schr{ö}dinger equation, it is the first time that the derivation of an error estimate, for a fully discrete method based on the Relaxation Scheme, is completely addressed.
