Showing papers in "Computers & Mathematics With Applications in 2016"

Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma

Abstract: The nonlinear three-dimensional modified Zakharov-Kuznetsov (mZK) equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in a presence of a uniform magnetic field. By using the reductive perturbation procedure leads to a mZK equation governing the propagation of ion dynamics of nonlinear ion-acoustic waves in a plasma. The mZK equation has solutions that represent solitary traveling waves. We found the electrostatic field potential and electric field in form traveling wave solutions for three-dimensional mZK equation. The solutions for the mZK equation are obtained precisely and efficiency of the method can be demonstrated. The stability of solitary traveling wave solutions of the mZK equation to three-dimensional long-wavelength perturbations are investigated.

Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations

Abstract: Lump solitons are obtained from (2+1)-dimensional Ito equation and (2+1)-dimensional CDGKS equation by using a direct method and the completely non-elastic interaction between a lump and a stripe of (2+1)-dimensional Ito equation is presented, which shows a lump soliton is drowned or swallowed by a stripe soliton. The direct method is based on Hirota bilinear form of the equations that have special solutions of quadratic function. The general formula of coordinates of the vertex and the amplitude of lump solitons are given. The dynamical character shows in this work enriches the variety of the dynamics of higher dimensional nonlinear wave field.

Resonant behavior of multiple wave solutions to a Hirota bilinear equation

Abstract: For the exponential traveling wave solutions to the Hirota bilinear equations, a sufficient and necessary criterion for the existence of linear superposition principle has been given. Motivated by this criterion, we propose a new Hirota bilinear equation via using a multivariate polynomial. Applying the linear superposition principle to this new Hirota bilinear equation, we finally find two types of resonant multiple wave solutions, among which, the resonant three-wave solutions are illustrated with three-dimensional plots.

Breaking spaces and forms for the DPG method and applications including Maxwell equations

Abstract: Discontinuous Petrov-Galerkin (DPG) methods are made easily implementable using "broken" test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and various forms in between.

Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation

Abstract: Associated with the prime number p = 3 , the generalized bilinear operators are adopted to yield an extended Kadomtsev-Petviashvili-like (eKP-like) equation. With symbolic computation, eighteen classes of rational solutions to the resulting eKP-like equation are generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

Convergence studies in meshfree peridynamic simulations

Abstract: Meshfree methods are commonly applied to discretize peridynamic models, particularly in numerical simulations of engineering problems. Such methods discretize peridynamic bodies using a set of nodes with characteristic volume, leading to particle-based descriptions of systems. In this paper, we perform convergence studies of static peridynamic problems. We show that commonly used meshfree methods in peridynamics suffer from accuracy and convergence issues, due to a rough approximation of the contribution of nodes near the boundary of the neighborhood of a given node to numerical integrations. We propose two methods to improve meshfree peridynamic simulations. The first method uses accurate computations of volumes of intersections between neighbor cells and the neighborhood of a given node, referred to as partial volumes. The second method employs smooth influence functions with a finite support within peridynamic kernels. Numerical results demonstrate great improvements in accuracy and convergence of peridynamic numerical solutions when using the proposed methods.

Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation

Abstract: We propose three Fourier spectral methods, i.e.,źthe split-step Fourier spectral (SSFS), the Crank-Nicolson Fourier spectral (CNFS), and the relaxation Fourier spectral (ReFS) methods, for solving the fractional nonlinear Schrodinger (NLS) equation. All of them are mass conservative and time reversible, and they have the spectral order accuracy in space and the second-order accuracy in time. In addition, the CNFS and ReFS methods are energy conservative. The performance of these methods in simulating the plane wave and soliton dynamics is discussed. The SSFS method preserves the dispersion relation, and thus it is more accurate for studying the long-time behaviors of the plane wave solutions. Furthermore, our numerical simulations suggest that the SSFS method is better in solving the defocusing NLS, but the CNFS and ReFS methods are more effective for the focusing NLS.

Radial basis function partition of unity methods for pricing vanilla basket options

Abstract: Meshfree methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A?disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity. RBF partition of unity methods (RBF-PUM) allow for a significant sparsification of the linear system and lower the computational effort. In this work we apply a global RBF method as well as RBF-PUM to problems in option pricing. We consider one- and two-dimensional vanilla options. In order to price American options we employ a penalty approach. A penalty term, suitable for American multi-asset call options, has been designed. RBF-PUM is shown to be competitive compared with a finite difference method and a global RBF method. It is as accurate as the global RBF method, but significantly faster. The results for RBF-PUM look promising for extension to higher-dimensional problems.

Free vibration of functionally graded carbon nanotube reinforced composite plates integrated with piezoelectric layers

Abstract: In the present research, free vibration behavior of carbon nanotube reinforced composite (CNTRC) plates integrated with piezoelectric layers at the bottom and top surfaces is analyzed. Plate is modeled according to the first order shear deformation plate theory. Distribution of CNTs across the plate thickness may be functionally graded (FG) or uniformly distributed (UD). Properties of the composite media are obtained according to a modified rule of mixtures approach which contains efficiency parameters. Distribution of electric potential across the piezoelectric thickness is assumed to be linear. The complete set of motion and Maxwell equations of the system are obtained according to the Ritz formulation suitable for arbitrary in-plane and out-of-plane boundary conditions. Besides, two types of electrical boundary conditions, namely, closed circuit and open circuit are considered for the free surfaces of the piezoelectric layers. Chebyshev polynomials are used as the basis functions in Ritz approximation. The resultant eigenvalue system is solved to obtain the frequencies of the system as well as the mode shapes. It is shown that, fundamental frequency of a closed circuit plate is always higher than a plate with open circuit boundary conditions.

A new design for the implementation of isogeometric analysis in Octave and Matlab

Abstract: GeoPDEs ź(http://rafavzqz.github.io/geopdes) is an Octave/Matlab package for the solution of partial differential equations with isogeometric analysis, first released in 2010. In this work we present in detail the new design of the package, based on the use of Octave and Matlab classes. Compared to previous versions the new design is much clearer, and it is also more efficient in terms of memory consumption and computational time.

Numerical solutions for gyrotactic bioconvection in nanofluid-saturated porous media with Stefan blowing and multiple slip effects

Abstract: A mathematical model is developed to examine the effects of the Stefan blowing, second order velocity slip, thermal slip and microorganism species slip on nonlinear bioconvection boundary layer flow of a nanofluid over a horizontal plate embedded in a porous medium with the presence of passively controlled boundary condition. Scaling group transformations are used to find similarity equations of such nanobioconvection flows. The similarity equations are numerically solved with a Chebyshev collocation method. Validation of solutions is conducted with a Nakamura tri-diagonal finite difference algorithm. The effects of nanofluid characteristics and boundary properties such as the slips, Stefan blowing, Brownian motion and Grashof number on the dimensionless fluid velocity, temperature, nanoparticle volume fraction, motile microorganism, skin friction, the rate of heat transfer and the rate of motile microorganism transfer are investigated. This work is relevant to bio-inspired nanofluid-enhanced fuel cells and nano-materials fabrication processes.

Numerical solution of the time fractional Black-Scholes model governing European options

Abstract: When considering the price change of the underlying fractal transmission system, a fractional Black-Scholes(B-S) model with an α -order time fractional derivative is derived. In this paper, we discuss the numerical simulation of this time fractional Black-Scholes model (TFBSM) governing European options. A discrete implicit numerical scheme with a spatially second-order accuracy and a temporally 2 - α order accuracy is constructed. Then, the stability and convergence of the proposed numerical scheme are analyzed using Fourier analysis. Some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method. Finally, as an application, we use the TFBSM and the above numerical technique to price several different European options.

On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics

Abstract: In this paper, a ( 3 + 1 ) -dimensional generalized B-type Kadomtsev–Petviashvili equation is investigated, which can be used to describe weakly dispersive waves propagating in a quasi media and fluid mechanics. Based on the Bell polynomials, its multiple-soliton solutions and the bilinear form with some reductions are derived, respectively. Furthermore, by using Riemann theta function, we construct one- and two-periodic wave solutions for the equation. Finally, we study the asymptotic behavior of the periodic wave solutions, which implies that the periodic wave solutions can be degenerated to the soliton solutions under a small amplitude limit.

On the stability of the moving least squares approximation and the element-free Galerkin method

Abstract: In this paper, the stability of the moving least squares (MLS) approximation and a stabilized MLS approximation is analyzed theoretically and verified numerically. It is shown that the stability of the MLS approximation deteriorates severely as the nodal spacing decreases, while the stability of the stabilized MLS approximation is not affected by the nodal spacing. The stabilized MLS approximation is introduced into the element-free Galerkin (EFG) method to produce a stabilized EFG method. Theoretical error analysis of the stabilized EFG method is provided for boundary value problems with mixed boundary conditions of Dirichlet and Robin type. Numerical examples confirm the theoretical results, and show that the stabilized EFG method has higher computational precision and better stability than the original EFG method.

Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions

Abstract: In this work, a meshless scheme is presented for the fast simulation of multi-dimensional wave problems. The present method is rather simple and straightforward. The Houbolt method is used to eliminate the time dependence of spatial variables. Then the original wave problem is converted into equivalent systems of modified Helmholtz equations. The sparse scheme of the method of fundamental solutions in combination with the localized method of approximate particular solutions is employed for efficient implementation of spatial variables. To demonstrate the effectiveness and simplicity of this new approach, three numerical examples have been assessed with excellent performance.

Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method

Abstract: This short communication presents the Singular Boundary Method (SBM) to solve the obliquely incident water wave passing through the submerged breakwater. The SBM is a recent boundary-type collocation method with the merits of being meshless, integration-free, mathematically simple, easy-to-program, well suited for unbounded domain problems. The accuracy and efficiency of the SBM is first verified in several benchmark examples in comparison with the boundary element method. Then the effects of the position, size and geometry of the breakwater on water wave propagation are investigated through numerical experiments, and some new observations are concluded.

Finite element approximation of optimal control problems governed by time fractional diffusion equation

Abstract: In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated. Piecewise linear polynomials are used to approximate the state and adjoint state, while the control is discretized by variational discretization method. A priori error estimates for the semi-discrete approximations of the state, adjoint state and control are derived. Furthermore, we also discuss the fully discrete scheme for the control problems. A finite difference method developed in Lin and Xu (2007) is used to discretize the time fractional derivative. Fully discrete first order optimality condition is developed based on 'first discretize, then optimize' approach. The stability and truncation error of the fully discrete scheme are analyzed. Numerical example is given to illustrate the theoretical findings.

Implementation issues and benchmarking of lattice Boltzmann method for moving rigid particle simulations in a viscous flow

Abstract: In this work, we revisit implementation issues in the lattice Boltzmann method (LBM) concerning moving rigid solid particles suspended a viscous fluid. Three aspects relevant to the interaction between flow of a viscous fluid and moving solid boundaries are considered. First, the popular interpolated bounce back scheme is examined both theoretically and numerically. It is important to recognize that even though significant efforts had previously been devoted to the performance, especially the accuracy, of different interpolated bounce back schemes for a fixed boundary, there were relatively few studies focusing on moving solid surfaces. In this study, different interpolated bounce back schemes are compared theoretically for a moving boundary. Then, several benchmark cases are presented to show their actual performance in numerical simulations. Second, we examine different implementations of the momentum exchange method to calculate hydrodynamic force and torque acting on a moving surface. The momentum exchange method is well established for fixed solid boundaries, however, for moving solid boundaries there are still open issues such as unphysical force fluctuations and Galilean invariance errors. Recent progress in this direction is discussed, along with our own interpretations and modifications. Several benchmark cases, including a particle-laden turbulent channel flow, are used to demonstrate the effects of different modifications on the accuracy and physical results under different physical configurations. The third aspect is the refilling scheme for constructing the unknown distribution functions for the new fluid nodes that emerge from the previous solid region as a particle moves relative to a fixed lattice grid. We examine and compare the performance of the refilling schemes introduced by Fang etźal. (2002), Lallemand and Luo (2003), and Caiazzo (2008). We demonstrate that improvements can be made to suppress force fluctuations resulting from refilling.

Adjoint Lattice Boltzmann for topology optimization on multi-GPU architecture

Abstract: In this paper we present a topology optimization technique applicable to a broad range of flow design problems. We propose also a discrete adjoint formulation effective for a wide class of Lattice Boltzmann Methods (LBM). This adjoint formulation is used to calculate sensitivity of the LBM solution to several type of parameters, both global and local. The numerical scheme for solving the adjoint problem has many properties of the original system, including locality and explicit time-stepping. Thus it is possible to integrate it with the standard LBM solver, allowing for straightforward and efficient parallelization (overcoming limitations typical for the discrete adjoint solvers). This approach is successfully used for the channel flow to design a free-topology mixer and a heat exchanger. Both resulting geometries being very complex maximize their objective functions, while keeping viscous losses at acceptable level.

Analytic regularity and collocation approximation for elliptic PDEs with Random domain deformations

Abstract: In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped onto a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in C^N with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and variance of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.

Virtual Element Method for fourth order problems: $L^2-$estimates

Abstract: We analyse the family of C 1 -Virtual Elements introduced in Brezzi and Marini (2013) for fourth-order problems and prove optimal estimates in L 2 and in H 1 via classical duality arguments.

Geometry models of porous media based on Voronoi tessellations and their porosity-permeability relations

Abstract: In this paper, we present methods that directly model the random structure of porous media using Voronoi tessellations. Three basic structures were generated and they correspond to porous medium geometries with intersecting fractures (granular), interconnected tubes (tubular), and fibers (fibrous). Fluid flow through these models was solved by a massively parallelized lattice Boltzmann code. We established the porosity-permeability relations for these basic geometry models. It is found that, for granular and tubular geometries, the specific surface area is a critical structural parameter that can bring their porosity-permeability relations together under a unified Kozeny-Carman equation. A connected fracture network, superimposed on the basic Voronoi structure, increases the dimensionless permeability relative to the Kozeny-Carman equation; isolated large pores (vugs), on the other hand, decreases the dimensionless permeability relative to the Kozeny-Carman equation. The Kozeny-Carman equation, however, cannot distinguish a heterogeneous structure with an embedded partially penetrating fracture. The porosity-permeability relation for fibrous geometries in general agrees with those established for simple-cubic, body-centered cubic, and face-centered cubic models. In the dilute limit, however, the dependence on the solid fraction is weaker in Voronoi geometries, indicating weaker hydrodynamic interactions among randomly interconnected fibers than those in the idealized models.

Parallel coupling numerics for partitioned fluid–structure interaction simulations

Abstract: Within the last decade, very sophisticated numerical methods for the iterative and partitioned solution of fluid–structure interaction problems have been developed that allow for high accuracy and very complex scenarios. The combination of these two aspects–accuracy and complexity–demands very high computational grid resolutions and, thus, high performance computing methods designed for massively parallel hardware architectures. For those architectures, currently used coupling methods, which mainly work with a staggered execution of the fluid and the structure solver, i.e., the execution of one solver after the other in every outer iteration, lead to severe load imbalances: if the flow solver, e.g., scales on a very large number of processors but the structural solver does not due to its limited amount of data and required operations, almost all processors assigned to the coupled simulations are idle during the execution of the structure solver. We propose two new iterative coupling methods that allow for the simultaneous execution of flow and structure solvers. In both cases, we show that pure fixed-point iterations based on the parallel execution of the solvers do not lead to good results, but the combination of parallel solver execution and so-called quasi-Newton methods yields very efficient and robust methods. Those methods are known to be very efficient also for the stabilization of critical scenarios solved with the standard staggered solver execution. We demonstrate the competitive convergence of our methods for various established benchmark scenarios. Both methods are perfectly suited for use with black-box solvers because the quasi-Newton approach uses solely input and output information of the solvers to approximate the effect of the unknown Jacobians that would be required in a standard Newton solver.

A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions

Abstract: We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. When some assumptions on the kernel functions hold, we prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia's agile software components toolkit. The latter provides the groundwork for the development of engineering analysis tools, while numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method.

The generalized ( 1 + 1 )-dimensional and ( 2 + 1 )-dimensional Ito equations

Abstract: The multiple exp-function algorithm, as a generalization of Hirota's perturbation scheme, is used to construct multiple wave solutions to the generalized ( 1 + 1 )-dimensional and ( 2 + 1 )-dimensional Ito equations. Some of the resulting solutions involve generic phase shifts and wave frequencies.

Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach

Abstract: In this work, the symmetry group and similarity reductions of the two-dimensional generalized Benney system are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Firstly, the vector field associated with the Lie group of transformation is obtained. Then the point transformations are proposed, which keep the solutions of the generalized Benney system invariant. Finally, the symmetry reductions and explicitly exact solutions of the generalized Benney system are derived by solving the corresponding symmetry equations.

Different methods for ( 3 + 1 ) -dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation

Abstract: In this paper, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the exp-function method, the ( G ' G ) -expansion method and the generalized Kudryashov method?are used to construct exact solutions for ( 3 + 1 ) -dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation. This fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it. As a result, some new exact solutions are obtained. The three methods demonstrate power, reliability and efficiency.

Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay

Abstract: In this paper, we study the mild solutions of a class of nonlinear fractional reaction–diffusion equations with delay and Caputo’s fractional derivatives. By the measure of noncompactness, the theory of resolvent operators, the fixed point theorem and the Banach contraction mapping principle, we develop various theorems for the existence and uniqueness of the global mild solutions for the problem.

Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

Abstract: The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these "functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed. (C) 2015 Elsevier Ltd. All rights reserved.

New exact solutions of nonlinear fractional acoustic wave equations in ultrasound

Abstract: In this paper, new exact solutions of fractional nonlinear acoustic wave equations have been devised. The travelling periodic wave solutions of fractional Burgers-Hopf equation and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation have obtained by first integral method. Nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible. Taking nonlinear effects into account, the ultrasound beam analysis makes more accurate in these applications. The Burgers-Hopf equation is one of the extensively studied models in mathematical physics. In addition, the KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In the present analysis, these nonlinear equations have solved by first integral method. As a result, new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations. The obtained results are presented graphically to demonstrate the efficiency of this proposed method.