Two-dimensional Unbounded Domains Research Articles

A multi-domain spectral collocation method for the Fokker–Planck equation in an infinite channel

In this paper, we propose a multi-domain spectral collocation method for partial differential equations on two-dimensional unbounded domains. Some approximation results on the composite generalized Laguerre-Legendre interpolation and quasi-orthogonal projecting are established, respectively. These results play a significant role in related spectral collocation method. As an application, a multi-domain spectral collocation scheme is provided for the Fokker–Planck equation with absorption or non-homogeneous boundary conditions. The convergence of the proposed algorithm is performed. An efficient implementation is presented. Numerical experiments demonstrate the effectiveness and high accuracy of the algorithm.

Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions

The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. Our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.

Ergodicity for the Randomly Forced Navier–Stokes System in a Two-Dimensional Unbounded Domain

The ergodic properties of the randomly forced Navier–Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier–Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020) and Shirikyan (J Eur Math Soc, 2020) and using the asymptotic compactness of the dynamics.

Primal-dual active-set method for solving the unilateral pricing problem of American better-of options on two assets

<abstract><p>In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end.</p></abstract>

ON A BOUNDARY VALUE PROBLEM FOR ONE OVERDETERMINED SYSTEM IN A HALF-PLANE

This paper considers a boundary value problem for an overdetermined system of equations in a half-plane. This problem arises in particular when solving a stationary system of the two-velocity hydrodynamics with one pressure and homogeneous divergent conditions and the Dirichlet boundary conditions for two phase velocities, as well as in problems of electrodynamics. The generalized solution to a stationary system of the two-velocity hydrodynamics in the case of two-dimensional unbounded domains, for instance, in a half-plane, has a significant difference from the three-dimensional case. Namely, in the two-dimensional case for the velocities it is impossible to satisfy the pre-set conditions at infinity and the condition of boundedness at infinity is imposed. In this case, the medium is considered to be homogeneous, and the energy dissipation occurs due to the shear viscosities of the phases of the subsystems, and other effects are not discussed in this paper. The mass transfer occurs due to the mass force. With an appropriate choice of functional spaces, the existence and uniqueness of a generalized solution with an appropriate stability estimate has been proved.

Efficient Solution of Two-Dimensional Wave Propagation Problems by CQ-Wavelet BEM: Algorithm and Applications

In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their s...

Steady Navier-Stokes equations with Poiseuille and Jeffery-Hamel flows in $\mathbb R^2$

We consider a steady flow of an incompressible viscous fluid in a two-dimensional unbounded domain with unbounded boundaries. The domain has two outlets. The part of the domain upstream is a cylinder and the part of the domain downstream is a wedge. In the part of the domain upstream, the velocity is required to approach the Poiseuille flow. In the part of the domain downstream, the velocity is required to approach Jeffery-Hamel's flow. This problem has been treated by C.J. Amick and L.E. Frankel, V.A. Solonnikov and many others. Recently, T. Kobayashi obtained the unique solution of Jeffery-Hamel's flows in a wedge. Therefore we reconsider this problem. In this paper, we succeed in proving the existence of such a steady flow under the restricted flux condition which depends only on the part of the domain upstream and downstream.

Primal-dual active set method for pricing American better-of option on two assets

An American better-of option is a rainbow option with two underlying assets. It can be described by a parabolic variational inequality (VI) on a two-dimensional unbounded domain, which can also be characterized by a two-dimensional free boundary problem. Based on the numeraire transformation and the known information on the free boundary, we derive a one-dimensional linear complementarity problem (LCP) related to options on a bounded domain. Moreover, the full discretization scheme of LCP is constructed by finite difference and finite element methods in temporal and spatial directions, respectively. The primal-dual active-set (PDAS) method is adopted for solving the resulting large-scale discretized system. In each PDAS iteration, a unique index set of primal-dual variables will be computed by solving a linear system. A systematical convergent analysis is presented on our proposed method for pricing American better-of options. One of the desirable features of our method is that we can get the option value and two free boundaries simultaneously. Numerical simulations are performed to verify the efficiency of our proposed method.

ON THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION

In the paper it is considered the regularization of the Cauchy problem for systems of elliptic type equations of the first order with constant coefficients factorisable Helmholtz operator in two-dimensional unbounded domain. Using the results of [20, 21, 22, 23, 24, 25 ], we construct in explicit form Carleman matrix and, based on the regularized solution of the Cauchy problem.

A reduced-order extrapolated natural boundary element method based on POD for the parabolic equation in the 2D unbounded domain

In this article, we devote ourselves to the research of order reduction of natural boundary element (NBE) based on a proper orthogonal decomposition (POD) for the parabolic equation in the two-dimensional (2D) unbounded domain. For this purpose, we first build a NBE format for the parabolic equation in the 2D unbounded domain and discuss the existence, stability, and convergence of the NBE solutions. And then, we build a reduced-order NBE extrapolated (RONBEE) format based on POD, analyze the errors between the classical NBE and RONBEE solutions, and supply the implementation procedure for the RONBEE format. Finally, we utilize some numerical experiments to validate that the numerical computational consequences are consistent with the theoretical ones such that the effectiveness and feasibility of the RONBEE format are further verified.

Explicit/implicit multi-time step co-simulation in unbounded medium with Rayleigh damping and application for wave barrier

Co-simulation strategies using both Abaqus/Explicit and Abaqus/Implicit are investigated for analysing wave propagation in two-dimensional unbounded soil domain. The co-simulation is based on the coupling GC method, allowing for coupling different finite element codes with different time integrators and time-scales depending on the partitions of the domain. Absorbing layers using increasing damping (ALID), based on Rayleigh viscous damping, are considered at the boundary to model the semi-infinite medium. The proposed absorbing region is called hybrid (different time integrators) asynchronous (different time steps) absorbing layers using increasing damping (HA-ALID). Lamb’s test and a wave barrier problem are investigated. First, HA-ALID turns out to be more accurate than non-reflective conditions available in Abaqus/Explicit. Second, the critical time step in the domain of interest remains unaffected by the choice of damping characteristics in the HA-ALID, contrary to the case of a full explicit computation. Third, the time steps for the solid barrier and the HA-ALID are not restrained by the CFL condition imposed by the explicit partition for stability reasons. Because of its good accuracy and ability to be realized using only FE Abaqus package, without a third-party software component, this strategy provides a wide range of applications in soil–structure interaction problems.

Solenoidal extensions of vector fields in two-dimensional unbounded domains

The goal of this note is to construct solenoidal extensions of vector fields defined on the boundary of simply connected domains having outlets to infinity and which satisfy the Leray–Hopf condition. The case of non-simply connected domains is also mentioned, in particular in the case when the domain admits a symmetry axis. This kind of extensions allows us to solve the stationary Navier–Stokes problem with nonhomogeneous boundary conditions in such domains.

Hamiltonian-Based Derivation of the High-Performance Scaled-Boundary Finite-Element Method Applied to the Complex Multilayered Soil Field in Time Domain

AbstractWave propagation in a two-dimensional unbounded domain with rigid bedrock was studied. The study was based on the scaled-boundary finite-element method (SBFEM) combing the continued fraction method. The SBFEM is a novel semianalytical technique, combining the advantages of the finite-element and the boundary-element methods. Meanwhile, SBFEM has unique properties of its own. In this paper, the modified SBFEM was introduced. The original scaling center was replaced by a scaling line, which is more suitable for analyzing the multilayered soil model. As a key point, the modified SBFEM and the dual system, which is used to solve a proposed time domain problem, were combined to derive the governing equations. The derivation process was based on the framework of the Hamiltonian system by introducing dual variables. By using the continued fraction solution and introducing auxiliary variables, the equation of motion was built. More significantly, the efficient, precise time-integration method was first us...

A time domain solution for complex multilayered soil model with circular inhomogeneity by the SBFEM

The scaled boundary finite method (SBFEM) is developed for analyzing wave propagation problem in the two-dimensional unbounded domain with rigid bedrock. It combines the advantages of the finite element method and boundary element boundary method. Moreover, the original scaling center is replaced by a scaling line which is more suitable for analyzing the multilayered soil model. Therefore, the modified SBFEM develops the original SBFEM. A new derivation of the modified SBFEM equation is built in the frame of Hamilton system. A continued fraction solution of the dynamic stiffness of the soil model with bedrock is obtained for the first time. Then, by introducing the continued fraction solution and auxiliary variables, it leads to the model resting on bedrock can be solved in time domain. The global equation of motion is solved by the efficient precise time-integration method. This integral method is firstly employed in the modified SBFEM. The precision of the proposed method can achieve computer precision. Therefore, an extremely efficient and accurate solution of the modified SBFEM in time domain is obtained. The results of the complex soil model with circular inhomogeneity show that the proposed method yields excellent results, and high accuracy is observed.

Time-domain analyses of the layered soil by the modified scaled boundary finite element method

The dynamic response of two-dimensional unbounded domain on the rigid bedrock in the time domain is numerically obtained. It is realized by the modified scaled boundary finite element method (SBFEM) in which the original scaling center is replaced by a scaling line. The formulation bases on expanding dynamic stiffness by using the continued fraction approach. The solution converges rapidly over the whole time range along with the order of the continued fraction increases. In addition, the method is suitable for large scale systems. The numerical method is employed which is a combination of the time domain SBFEM for far field and the finite element method used for near field. By using the continued fraction solution and introducing auxiliary variables, the equation of motion of unbounded domain is built. Applying the spectral shifting technique, the virtual modes of motion equation are eliminated. Standard procedure in structural dynamic is directly applicable for time domain problem. Since the coefficient matrixes of equation are banded and symmetric, the equation can be solved efficiently by using the direct time domain integration method. Numerical examples demonstrate the increased robustness, accuracy and superiority of the proposed method. The suitability of proposed method for time domain simulations of complex systems is also demonstrated.

Computation of steady incompressible flows in unbounded domains

In this study we revisit the problem of computing steady Navier–Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction–vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background “skeleton” field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favorably against known theoretical predictions and the data available in the literature.

Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain

We consider the numerical solution of the time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain. Introduce an artificial boundary and find the exact and approximate artificial boundary conditions for the given problem, which lead to a bounded computational domain. Using the exact or approximating boundary conditions on the artificial boundary, the original problem is reduced to an initial–boundary-value problem on the bounded computational domain which is respectively equivalent to or approximates the original problem. A finite difference method is used to solve the reduced problems on the bounded computational domain. The numerical results demonstrate that the method given in this paper is effective and feasible.

Obtaining artificial boundary conditions for fractional sub-diffusion equation on space two-dimensional unbounded domains

In this paper, we consider fractional sub-diffusion equation on two-dimensional unbounded domains and obtain an exact and some approximating artificial boundary conditions for it by a joint application of the Laplace transform and the Fourier series expansion. In order to test the derived artificial boundary condition, after reducing the main problem to an initial–boundary value problem on bounded domain by imposing the obtained boundary condition, we just use classical Crank–Nicolson method for space variables and L1 approximation for the fractional time derivative. Some numerical examples are given which confirm the effectiveness of the proposed artificial boundary conditions.

Finite-dimensional Pullback Attractors for Non-autonomous Newton–Boussinesq Equations in Some Two-dimensional Unbounded Domains

Finite-dimensional Pullback Attractors for Non-autonomous Newton–Boussinesq Equations in Some Two-dimensional Unbounded Domains

New methodologies for the calculation of Green´s functions for wave problems in two-dimensional unbounded domains

This work describes the application of new methodologies for the evaluation of the inverse Fourier transforms that yield Green's functions for both the wave and Helmholtz equations in the entire bidimensional domain.