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We first prove the second order convergence of the Strang-type splitting scheme for the nonlinear Schrödinger equation. The proof does not require commutator estimates but crucially relies on an integral representation of the scheme. It reveals the connection between Strang-type splitting and the midpoint rule. We then show that the integral representation idea can also be used to study the stochastic nonlinear Schrödinger equation with multiplicative noise of Stratonovich type. Even though the nonlinear term there is not globally Lipschitz, we prove the first order convergence of a splitting scheme of it. Both schemes preserve the mass. They are very efficient because they use explicit formulas to solve the subproblems containing the nonlinear or the nonlinear plus stochastic terms.

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We present a class of semi-implicit finite element (FE) schemes that uses arbitrary Lagrangian Eulerian methods (ALE) to solve the incompressible Navier-Stokes equations (NSE) on time varying domains. We use the kth order backward differentiation formula (BDFk) and TaylorHood Pm/P m−1 finite elements. The well-known telescope formulas of BDFk have been extended from k = 1, 2 to k = 3, 4, 5. They enable us to prove that when k ≤ 5, for Stokes equations on a fixed domain, our schemes converge at rate O(Δt k + h m+1 ). When the domain is varying with respect to time and when h/Δt = O(1), the convergence rate reduces to O(Δt k + h m ). For analysis, we assume that meshes at different time levels have the same topology. Consequently, our methods do not require the computation of characteristic paths and are Jacobian-free. Numerical tests for NSE on time varying domains are presented. They indicate that our schemes may have full accuracy on time varying domains and can handle meshes with large aspect ratio. The benchmark test of flow past an oscillating cylinder is also performed.

In this article, a new regularization method is proposed and applied to the identification of multi-source dynamic loads acting on a surface of composite laminated cylindrical shell. In general terms, regularization is the approximation of an ill-posed problem by a family of neighbouring well-posed problems. Based on the construction of a new regularization operator, corresponding regularization method is established. We propose a regularization operator, thus construct a new regularization method, and prove its regular property. The present method is applied to load identification of composite laminated cylindrical shell. The transient displacement response can be obtained by the finite element method. The multi-source dynamic loads on a surface of composite laminated cylindrical shell are successfully identified, which demonstrates the efficiency and robustness of the present method.