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
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.
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.
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.
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 $t$.
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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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.
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.
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
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.
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.