An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem.     

Pseudospectral method for fractional infinite horizon optimal control problems

Up to now, several numerical methods have been presented to solve finite horizon fractional optimal control problems by researchers, while solving fractional optimal control problems on infinite domain is a challenging work. Hence, in this article, a numerical method is proposed to solve fractional infinite horizon optimal control problems. At the first stage, a domain transformation technique is used to map the infinite domain to a finite horizon. Also, fractional derivative defined on an unbounded domain is converted into an equivalent derivative on a finite domain. Then, a new shifted Legendre pseudospectral method is utilized to solve the obtained finite problem and a nonlinear programming problem is suggested to approximate the optimal solutions. Finally, some numerical examples are given to show the efficiency of the method.     

Analysis of High-Order Absorbing Boundary Conditions for the Schrödinger Equation

The paper is concerned with the numerical solution of Schrödinger equations on an unbounded spatial domain. High-order absorbing boundary conditions for one-dimensional domain are derived, and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate. Then a second order finite difference scheme is proposed, and the convergence of the scheme is established as well. Finally, numerical examples are reported to confirm our error estimates of the numerical methods.     

A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are defined by suitable level-set functions. The equations are discretized by classical nine-point stencil on interior points, while boundary conditions and high order reconstructions are used to define the field variables at ghost-points, which are grid nodes external to the domain with a neighbor inside the domain. The linear system arising from such discretization is solved by a multigrid strategy. The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources. The method is suitable to treat problems in which the geometry of the source often changes (explore the effects of different scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modified. Several numerical tests are successfully performed, which asses the effectiveness of the present approach.     

Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation

We develop fractional buffer layers (FBLs) to absorb propagating waves without reflection in bounded domains. Our formulation is based on variable-order spatial fractional derivatives. We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain. In particular, we first design proper FBLs for the one-dimensional one-way and two-way wave propagation. Then, we extend our formulation to two-dimensional problems, where we introduce a consistent variable-order fractional wave equation. In each case, we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time. We compare our results with a finely tuned perfectly matched layer (PML) method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular domain. We also demonstrate that our formulation is more robust and uses less number of equations.     

Split Local Artificial Boundary Conditions for the Two-Dimensional Sine-Gordon Equation on R2

In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied. Split local artificial boundary conditions are obtained by the operator splitting method. Then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method, and some interesting propagation and collision behaviors of the solitary wave solutions are observed.     

New Finite-Volume Relaxation Methods for the Third-Order Differential Equations

We propose a new method for numerical solution of the third-order differential equations. The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter. The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations. A non-oscillatory finite volume method for the relaxation system is developed. The method is uniformly accurate for all relaxation rates. Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation. Our method demonstrated the capability of accurately capturing soliton wave phenomena.     

Numerical Method for the Deterministic Kardar-Parisi-Zhang Equation in Unbounded Domains

We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-, two- and three- dimensional unbounded domains. The exact artificial boundary conditions are obtained on the artificial boundaries. Then the original problems are reduced to equivalent problems in bounded domains. A finite difference method is applied to solve the reduced problems, and some numerical examples are provided to show the effectiveness of the method.     

Wave Propagation Simulation Using the CIP Method of Characteristic Equations

We apply the CIP (Cubic Interpolated Profile) scheme to the numerical simulation of the acoustic wave propagation based on characteristic equations.The CIP scheme is based on a concept that both the wavefield and its spatial derivative propagate along the same characteristic curves derived from a hyperbolic differential equation. We describe the derivation of the characteristic equations for the acoustic waves from the basic equations by means of the directional splitting and the diagonalization of the coefficient matrix, and establish geophysical boundary conditions. Since the CIP scheme calculates both the wavefield and its spatial derivatives, it is easy to realize the boundary conditions theoretically. We also show some numerical simulation examples and the CIP can simulate acoustic wave propagation with high stability and less numerical dispersion. The method of characteristics with the CIP scheme is a very powerful technique to deal with the wave propagation in complex geophysical problems.     

Numerical Soliton Solutions for a Discrete Sine-Gordon System

In this paper we use an analytical-numerical approach to find, in a systematic way, new 1-soliton solutions for a discrete sine-Gordon system in one spatial dimension. Since the spatial domain is unbounded, the numerical scheme employed to generate these soliton solutions is based on the artificial boundary method. A large selection of numerical examples provides much insight into the possible shapes of these new 1-solitons.     

An Adaptive Finite Element PML Method for the Open Cavity Scattering Problems

Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the perfectly matched layer (PML) technique is developed to solve the reduced problem. The estimate takes account ofboththe finite element approximation error and the PML truncation error, where the latter is shown to decay exponentially with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the proposed method.     

Optimal spatial field control for controlled release

Most distributed parameter control problems involve manipulation within the spatial domain. Such problems arise in a variety of applications including epidemiology, tissue engineering, and cancer treatment. This paper proposes an approach to solve a state‐constrained spatial field control problem that is motivated by a biomedical application. In particular, the considered manipulation over a spatial field is described by partial differential equations (PDEs) with spatial frequency constraints. The proposed optimization algorithm for tracking a reference spatial field combines three‐dimensional Fourier series, which are truncated to satisfy the spatial frequency constraints, with exploitation of structural characteristics of the PDEs. The computational efficiency and performance of the optimization algorithm are demonstrated in a numerical example. In the example, the spatial tracking error is shown to be almost entirely due to the limitation on the spatial frequency of the manipulated field. The numerical results suggest that the proposed optimal control approach has promise for controlling the release of macromolecules in tissue engineering applications.     

GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforth scheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.     

Numerical Methods for Two-Fluid Dispersive Fast MHD Phenomena

The finite volume wave propagation method and the finite element RungeKutta discontinuous Galerkin (RKDG) method are studied for applications to balance laws describing plasma fluids. The plasma fluid equations explored are dispersive and not dissipative. The physical dispersion introduced through the source terms leads to the wide variety of plasma waves. The dispersive nature of the plasma fluid equations explored separates the work in this paper from previous publications. The linearized Euler equations with dispersive source terms are used as a model equation system to compare the wave propagation and RKDG methods. The numerical methods are then studied for applications of the full two-fluid plasma equations. The two-fluid equations describe the self-consistent evolution of electron and ion fluids in the presence of electromagnetic fields. It is found that the wave propagation method, when run at a CFL number of 1, is more accurate for equation systems that do not have disparate characteristic speeds. However, if the oscillation frequency is large compared to the frequency of information propagation, source splitting in the wave propagation method may cause phase errors. The Runge-Kutta discontinuous Galerkin method provides more accurate results for problems near steady-state as well as problems with disparate characteristic speeds when using higher spatial orders.     

A Multilevel Numerical Approach with Application in Time-Domain Electromagnetics

An algebraic multilevel method is proposed for efficiently simulating linear wave propagation using higher-order numerical schemes. This method is used in conjunction with the Finite Volume Time Domain (FVTD) technique for the numerical solution of the time-domain Maxwell's equations in electromagnetic scattering problems. In the multilevel method the solution is cycled through spatial operators of varying orders of accuracy, while maintaining highest-order accuracy at coarser approximation levels through the use of the relative truncation error as a forcing function. Higher-order spatial accuracy can be enforced using the multilevel method at a fraction of the computational cost incurred in a conventional higher-order implementation. The multilevel method is targeted towards electromagnetic scattering problems at large electrical sizes which usually require long simulation times due to the use of very fine meshes dictated by point-per-wavelength requirements to accurately model wave propagation over long distances.     

Weak Galerkin Method for Second-Order Elliptic Equations with Newton Boundary Condition

The weak Galerkin (WG) method is a nonconforming numerical method for solving partial differential equations. In this paper, we introduce the WG method for elliptic equations with Newton boundary condition in bounded domains. The Newton boundary condition is a nonlinear boundary condition arising from science and engineering applications. We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions. The error estimates are derived. Numerical experiments are presented to verify the theoretical analysis.     

Optimal shape design and unilateral boundary value problems: Part I

In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included.     

Layer-Splitting Methods for Time-Dependent Schrödinger Equations of Incommensurate Systems

This work considers numerical methods for the time-dependent Schrödinger equation of incommensurate systems. By using a plane wave method for spatial discretization, the incommensurate problem is lifted to a higher dimension that results in semidiscrete differential equations with extremely demanding computational cost. We propose several fully discrete time stepping schemes based on the idea of "layer-splitting", which decompose the semidiscrete problem into sub-problems that each corresponds to one of the periodic layers. Then these schemes handle only some periodic systems in the original lower dimension at each time step, which reduces the computational cost significantly and is natural to involve stochastic methods and parallel computing. Both theoretical analysis and numerical experiments are provided to support the reliability and efficiency of the algorithms.     

Finding initial costates in finite‐horizon nonlinear‐quadratic optimal control problems

A procedure for obtaining the initial value of the costate in a regular, finite‐horizon, nonlinear‐quadratic problem is devised in dimension one. The optimal control can then be constructed from the solution to the Hamiltonian equations, integrated on‐line. The initial costate is found by successively solving two first‐order, quasi‐linear, partial differential equations (PDEs), whose independent variables are the time‐horizon duration T and the final‐penalty coefficient S. These PDEs need to be integrated off‐line, the solution rendering not only the initial condition for the costate sought in the particular (T, S)‐situation but also additional information on the boundary values of the whole two‐parameter family of control problems, that can be used for design purposes. Results are tested against exact solutions of the PDEs for linear systems and also compared with numerical solutions of the bilinear‐quadratic problem obtained through a power‐series' expansion approach. Bilinear systems are specially treated in their character of universal approximations of nonlinear systems with bounded controls during finite time‐periods. Copyright © 2007 John Wiley & Sons, Ltd.