Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces
Xiaohai Wan Zhilin Li
Solving a Helmholtz equation $\Delta u + \lambda u = f$ efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of $\lambda$ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient $\lambda$ is inversely proportional to the mesh size.
keywords: augmented IIM. immersed interface method Helmholtz equation irregular domain discrete maximum principle elliptic interface problem finite difference method level set function embedding method
On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized $(F,\rho)$-convexity
Xiuhong Chen Zhihua Li

Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a non-differentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke sub-differentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the $G-(F, ρ)$ convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond-Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs.

keywords: Interval-valued function Clarke sub-differentiability G-(F, ρ) convexity optimality condition duality
An augmented immersed interface method for moving structures with mass
Jian Hao Zhilin Li Sharon R. Lubkin
We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.
keywords: Immersed interface method Navier-Stokes moving interface implicit scheme augmented method fluid-structure. projection method
Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay
Xiang Li Zhixiang Li
In this paper, we consider the long time behavior of a non-autonomous parabolic PDE with a discrete state-dependent delay. We study the existence of compact kernel sections and unique complete trajectory of the corresponding problem. Furthermore, we obtain the (almost) periodic solution which attracts all solutions provided the time dependent terms are (almost) periodic with respect to time $t$.
keywords: kernel section almost periodic solution. State-dependent delay periodic solution complete trajectory
Thomas P. Witelski David M. Ambrose Andrea Bertozzi Anita T. Layton Zhilin Li Michael L. Minion
Studies of problems in fluid dynamics have spurred research in many areas of mathematics, from rigorous analysis of nonlinear partial differential equations, to numerical analysis, to modeling and applied analysis of related physical systems. This special issue of Discrete and Continuous Dynamical Systems Series B is dedicated to our friend and colleague Tom Beale in recognition of his important contributions to these areas.

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Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes
Kai Liu Zhi Li
In this paper, we are concerned with a class of neutral stochastic partial differential equations driven by $\alpha$-stable processes. By combining some stochastic analysis techniques, tools from semigroup theory and delay integral inequalities, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the $p$-th moment to the stochastic systems are obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to the stochastic systems under consideration. Last, an example is presented to illustrate our theory in the work.
keywords: Global attracting set exponential decay in the $p$-th moment $\alpha$-stable process. stability in distribution
The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula
Zhiming Li Lin Shu
Consider a random cocycle $\Phi$ on a separable infinite-dimensional Hilbert space preserving a probability measure $\mu$, which is supported on a random compact set $K$. We show that if $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrable conditions of the differentials, then Pesin's entropy formula holds if $\mu$ has absolutely continuous conditional measures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.
keywords: Hilbert spaces Entropy formula infinite-dimensional random dynamical systems and SRB property.
The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation
Yila Bai Haiqing Zhao Xu Zhang Enmin Feng Zhijun Li
For the Arctic ice layer with high temperature in summer, the concepts of enthalpy degree, specific enthalpy and enthalpy conduction coefficient etc. are introduced in terms of the concept of enthalpy in calorifics. Heat conduction equation of enthalpy is constructed in the process of phase transformation of the Arctic ice. The condition of determinant solution and the identification model of diffusion coefficient of enthalpy are presented. Half implicit difference scheme and Schwartz alternating direction iteration are applied to solve the enthalpy conduction equation and sensitivity equation. Furthermore the Newton-Raphson algorithm is used for identification. The numerical results illustrate that the mathematic model and optimization algorithm are precise and feasible by the data(2003.8)of the Second Chinese Arctic Research Expedition in situ.
keywords: sea ice the Arctic numerical simulation. parameter identification
Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake
Qinxi Bai Zhijun Li Lei Wang Bing Tan Enmin Feng

Dissolved oxygen (DO) is one of the main parameters to assess the quality of lake water. This study is intended to construct a parabolic distributed parameter system to describe the variation of DO under the ice, and identify the vertical exchange coefficient K of DO with the field data. Based on the existence and uniqueness of the weak solution of this system, the fixed solution problem of the parabolic equation is transformed into a parameter identification model, which takes K as the identification parameter, and the deviation of the simulated and measured DO as the performance index. We prove the existence of the optimal parameter of the identification model, and derive the first order optimality conditions. Finally, we construct the optimization algorithm, and have carried out numerical simulation. According to the measured DO data in Lake Valkea-Kotinen (Finland), it can be found that the orders of magnitude of the coefficient K varying from 10-6 to 10-1 m2 s-1, the calculated and measured DO values are in good agreement. Within this range of K values, the overall trends are very similar. In order to get better fitting, the formula needs to be adjusted based on microbial and chemical consumption rates of DO.

keywords: Dissolved oxygen parameter identification numerical simulation partial differential equation frozen lakes
Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system
Zhigang Ren Shan Guo Zhipeng Li Zongze Wu

In this paper, an adjoint-based optimization method is employed to estimate the unknown coefficients and states arising in an one-dimensional (1-D) magnetohydrodynamic (MHD) flow, whose dynamics can be modeled by a coupled partial differential equations (PDEs) under some suitable assumptions. In this model, the coefficients of the Reynolds number and initial conditions as well as state variables are supposed to be unknown and need to be estimated. We first employ the Lagrange multiplier method to connect the dynamics of the 1-D MHD system and the cost functional defined as the least square errors between the measurements in the experiment and the numerical simulation values. Then, we use the adjoint-based method to the augmented Lagrangian cost functional to get an adjoint coupled PDEs system, and the exact gradients of the defined cost functional with respect to these unknown parameters and initial states are further derived. The existed gradient-based optimization technique such as sequential quadratic programming (SQP) is employed for minimizing the cost functional in the optimization process. Finally, we illustrate the numerical examples to verify the effectiveness of our adjoint-based estimation approach.

keywords: Magnetohydrodynamic parameter estimation state estimation adjoint method inverse problem data assimilation

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