Abstract: Publication date: Available online 14 February 2019Source: Applied Numerical MathematicsAuthor(s): Shuaichao Pei, Yanren Hou, Bo You In this paper, we propose a linear, unconditionally energy stable and second-order (in time) numerical scheme based on a convex splitting scheme and the semi-implicit spectral deferred correction (SISDC) method for the phase field crystal equation. The convex splitting scheme we use is linear, uniquely solvable and unconditionally energy stable but is of first-order, so we take the SISDC method to improve the rate of convergence. The resulted scheme inherits the advantages of the convex splitting scheme and thus leads to linear equations at each time step, which is easy to implement. We also prove that the scheme is unconditionally weak energy stable and of second-order accuracy in time. Numerical experiments are presented to validate the accuracy, efficiency and energy stability of the proposed numerical strategy.

Abstract: Publication date: Available online 12 February 2019Source: Applied Numerical MathematicsAuthor(s): J.E. Macías-Díaz In this work, we investigate numerically some relevant solutions of a diffusion problem with a general reaction term. The system under investigation is governed by a partial differential equation that generalizes the classical Fisher's equation and the Hodgkin–Huxley model. In a fist stage, we provide an equivalent presentation of the problem, and recall an explicit logarithmic numerical model to solve it. As one of the most important outcomes of this work, we prove that that logarithmic scheme is a structure-preserving model under suitable conditions. Concretely, we show that the numerical model is capable of preserving the positivity, the boundedness and the monotonicity of the approximations. This feature of the numerical integrator is in agreement with the fact that the continuous model has solutions with the same characteristics. Also, for the first time in the literature, we show that the logarithmic models are consistent schemes which are stable and convergent. Some numerical simulations confirm that the numerical model is a convergent technique, with first order of convergence in time and second order in space.

Abstract: Publication date: Available online 6 February 2019Source: Applied Numerical MathematicsAuthor(s): Hongwei Li, Xin Zhao, Yunxia Hu The numerical solution of the logarithmic Schrödinger equation on unbounded domains is considered in this paper. It is difficult to develop numerical methods for the logarithmic Schrödinger equation on unbounded domains, due to the blow up of the logarithmic nonlinearity and the unboundedness of the physical domain. Thus, a regularized version of the logarithmic Schrödinger equation on unbounded domains with a small regularization parameter is developed. Then, the local artificial boundary conditions for the regularized logarithmic Schrödinger equation are designed by applying the unified approach, which based on the idea of well-known operator splitting method. The regularized logarithmic Schrödinger equation defined on unbounded domains is reduced to an initial boundary value problem on the bounded computational domain, which can be solved by the finite difference method. The convergence and the stability of the reduced problem are analyzed by introducing some auxiliary variables. In order to choose the optimal absorb parameter in the local artificial boundary conditions, an adaptive algorithm is presented. Numerical results are reported to verify the accuracy and effectiveness of our proposed method.

Abstract: Publication date: Available online 4 February 2019Source: Applied Numerical MathematicsAuthor(s): Minam Moon, Raytcho Lazarov, Hyung Kyu Jun In this research, we give projection-based error analysis on a multiscale hybridizable discontinuous Galerkin method to numerically solve parabolic problem with a heterogeneous coefficient. We modified the spectral multiscale HDG method introduced in [25] to fit to the time-dependent PDE. The method uses multiscale spaces generated by eigenfunctions of local spectral problems. By considering two different grids, one relatively coarser than the other, we give bounds for the error between the actual solution and the approximate one derived from multiscale HDG method. One of the main focuses of the paper is to derive error analysis that depends on the size of fine and coarse grids and eigenvalues of local spectral problems. To solve a given coarse problem, the more eigenfunction we choose, the more accurate the approximation becomes: we shall see that the numerical result indicates that when we fix fine and coarse grids, the error between the reference solution and the derived one decreases whenever we have more multiscale basis functions.

Abstract: Publication date: Available online 31 January 2019Source: Applied Numerical MathematicsAuthor(s): N. Bazarra, M. Campo, J.R. Fernández, R. Quintanilla In this paper we study, from the numerical point of view, a thermoelastic problem with dual-phase-lag heat conduction. The variational formulation is written as a coupled system of hyperbolic linear variational equations. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability result is proved and a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.

Abstract: Publication date: Available online 30 January 2019Source: Applied Numerical MathematicsAuthor(s): Shishun Li, Rongliang Chen, Xinping Shao In this paper, we present a two-level space-time hybrid Schwarz preconditioner for GMRES based on the classical hybrid Schwarz method and the second-order backward differentiation formula. In the proposed method, the parabolic equations are solved in parallel on both of the space and time directions. Under some reasonable assumptions, the optimal convergence theory is developed for the proposed space-time method, i.e., the convergence rate is independent of the mesh parameters, the number of subdomains and the window size. Some numerical results are given to confirm the theory very well in terms of the convergence rate and accuracy. And the strong/weak scalability obtained with 4096 processors is also reported to show the efficiency of the proposed method.

Abstract: Publication date: Available online 30 January 2019Source: Applied Numerical MathematicsAuthor(s): Changpin Li, Zhen Wang In this article, three kinds of typical Caputo-type partial differential equations are numerically studied via the finite difference methods/the local discontinuous Galerkin finite element methods, including Caputo-type reaction-diffusion equation, Caputo-type reaction-diffusion-wave equation, and Caputo-type cable equation. The derived numerical schemes are unconditionally stable and convergent. The numerical experiments are also displayed which support the theoretical analysis.

Abstract: Publication date: Available online 28 January 2019Source: Applied Numerical MathematicsAuthor(s): Krister J. Trandal, Henrik Kalisch A long-wave model for the evolution of long waves at the interface of a deep and a shallow fluid is put forward. The model allows for a uniform stream in one of the layers, and the existence of interfacial capillarity. The model can be used to study the dynamics of the interface between liquid CO2 and seawater in the deep ocean, including the evolution of the hydrate layer.If restricted to unidirectional waves the model has the form of a Benjamin-type equation found by Benjamin [2]. Steady periodic solutions of the Benjamin equation are found using a numerical bifurcation code based on a pseudo-spectral projection. The bifurcation patterns are complex, with some branches featuring turning points and secondary bifurcations.

Abstract: Publication date: Available online 28 January 2019Source: Applied Numerical MathematicsAuthor(s): Shaobo Zhou, Hai Jin In the paper, our main aim is to investigate the strong convergence of the implicit numerical approximations for neutral-type stochastic differential equations with super-linearly growing coefficients. After providing mean-square moment boundedness and mean-square exponential stability for the exact solution, we show that a backward Euler-Maruyama approximation converges strongly to the true solution under polynomial growth conditions for sufficiently small step size. Imposing a few additional conditions, we examine the p-th moment uniform boundedness of the exact and approximate solutions by the stopping time technique, and establish the convergence rate of one half, which is the same as the convergence rate of the classical Euler-Maruyama scheme. Finally, several numerical simulations illustrate our main results.

Abstract: Publication date: Available online 28 January 2019Source: Applied Numerical MathematicsAuthor(s): Tinggan Yang, Yihong Wang The paper presents two Tailored Finite Point method (TFPM) for the diffusion equations, which is valid for the strongly anisotropic tensor diffusivity and interface layers. The first scheme uses the value as well as their derivatives at the grid points to construct the five point scheme for the heterogeneous rotating anisotropy. The second scheme is based on the interface conditions to construct the four point scheme for each cell, which gives rise to desirable internal layers and discontinuous diffusivity conditions. Numerically, both methods can achieve uniform convergence, even when there exhibit interface layers. Numerical experiments are presented to show the performance of the proposed two different schemes.

Abstract: Publication date: Available online 25 January 2019Source: Applied Numerical MathematicsAuthor(s): Dan Wu, Jingyan Yue, Guangwei Yuan, Junliang Lv Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods.

Abstract: Publication date: Available online 24 January 2019Source: Applied Numerical MathematicsAuthor(s): B.V. Rogov The Fourier analysis of fully discrete bicompact fourth-order spatial approximation schemes for hyperbolic equations is presented. This analysis is carried out on the example of a model linear advection equation. The results of Fourier analysis are presented as graphs of the dependence of the dispersion and dissipative characteristics of the bicompact schemes on the dimensionless wave number and the Courant number. The dispersion and dissipative properties of bicompact schemes are compared with those of other widely used difference schemes for hyperbolic equations. It is shown that bicompact schemes have one of the best spectral resolutions among the difference schemes being compared.

Abstract: Publication date: Available online 23 January 2019Source: Applied Numerical MathematicsAuthor(s): Angelamaria Cardone, Raffaele D'Ambrosio, Beatrice Paternoster The paper provides a spectral collocation numerical scheme for the approximation of the solutions of stochastic fractional differential equations. The discretization of the operator leads to a system of nonlinear algebraic equations, whose coefficient matrix can be computed by an automatic procedure, consisting of linear steps. A selection of numerical experiments confirming the effectiveness of the approach is given, with respect to various sets of function bases and of collocation points.

Abstract: Publication date: Available online 17 January 2019Source: Applied Numerical MathematicsAuthor(s): Yanbing Yang, Jiaheng Li, Tao Yu In this paper, we study the initial boundary value problem for Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term at three different initial energy levels, i.e. subcritical energy E(0)0. By potential well method, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases of E(0)0.

Abstract: Publication date: Available online 16 January 2019Source: Applied Numerical MathematicsAuthor(s): Helmi Temimi, Mohamed Ben-Romdhane, Mahboub Baccouch In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton-Raphson-Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ, where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme.

Abstract: Publication date: Available online 16 January 2019Source: Applied Numerical MathematicsAuthor(s): Hongfei Fu, Yanan Sun, Hong Wang, Xiangcheng Zheng We analyze a Crank-Nicolson finite volume method (CN-FVM) for the time-dependent two-sided conservative space fractional diffusion equation (of order 2−α). We prove that the proposed method is unconditionally stable in a weighted discrete norm and has a convergence rate of order O(τ2+h1+α), where τ and h are the time step size and spatial mesh size, respectively. In addition, we present a matrix-free preconditioned fast BiCGSTAB solver for the discrete linear algebraic system, which has a linear memory requirement and almost linear computational complexity. Numerical experiments show strong potential of the fast CN-FVM.

Abstract: Publication date: Available online 16 January 2019Source: Applied Numerical MathematicsAuthor(s): M.C. Calvo-Garrido, M. Ehrhardt, C. Vázquez In this paper we consider the valuation of swing options with the possibility of incorporating spikes in the underlying electricity price. This kind of contracts are modelled as path dependent options with multiple exercise rights. From the mathematical point of view the valuation of these products is posed as a sequence of free boundary problems where two consecutive exercise rights are separated by a time period. Due to the presence of jumps, the complementarity problems are associated with a partial-integro differential operator. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank-Nicolson semi-Lagrangian method for the time discretization of the differential part of the operator, jointly with the explicit treatment of the integral term by using the Adams-Bashforth scheme and combined with biquadratic Lagrange finite elements for space discretization. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature. Moreover, we employ appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results in order to illustrate the proper behaviour of the numerical schemes.

Abstract: Publication date: Available online 16 January 2019Source: Applied Numerical MathematicsAuthor(s): Yunzhang Zhang In this article, a first order linearized pressure correction projection method is proposed and analyzed for the time-dependent diffusive Peterlin viscoelastic model, which can describe the unsteady behavior of some incompressible polymeric fluids in two dimensions. Based on this method, the considered problem is decoupled into two subproblems: one is linearized for the velocity and conformation tensor, and the other is for the pressure. It is shown that the proposed projection scheme allows for a discrete energy inequality and is stable. In addition, we present a rigorous analysis for the convergence in time. Finally, some numerical examples are presented to verify the established theoretical findings.

Abstract: Publication date: Available online 15 January 2019Source: Applied Numerical MathematicsAuthor(s): Hu Chen, Finbarr Holland, Martin Stynes A convergence analysis is given for the Grünwald-Letnikov discretisation of a Riemann-Liouville fractional initial-value problem on a uniform mesh tm=mτ with m=0,1,…,M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t=0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grünwald-Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grünwald-Letnikov scheme is O(τtmα−1) at each mesh point tm; hence the scheme is globally only O(τα) accurate, but it is O(τ) accurate for mesh points tm that are bounded away from t=0. Numerical results for a test example show that these theoretical results are sharp.

Abstract: Publication date: Available online 11 January 2019Source: Applied Numerical MathematicsAuthor(s): Deepti Kaur, R.K. Mohanty In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto-Sivashinsky equation and the extended Fisher-Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

Abstract: Publication date: Available online 8 January 2019Source: Applied Numerical MathematicsAuthor(s): Shounian Deng, Weiyin Fei, Yong Liang, Xuerong Mao We present a stochastic age-dependent population model that accounts for Markovian switching and variable delay. By using the approximate value at the nearest grid-point on the left of the delayed argument to estimate the delay function, we propose a class of split-step θ-method for solving stochastic delay age-dependent population equations (SDAPEs) with Markovian switching. We show that the numerical method is convergent under the given conditions. Numerical examples are provided to illustrate our results.

Abstract: Publication date: Available online 4 January 2019Source: Applied Numerical MathematicsAuthor(s): Xiaoli Li, Hongxing Rui, Shuangshuang Chen In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes.

Abstract: Publication date: Available online 3 January 2019Source: Applied Numerical MathematicsAuthor(s): Mohammed Elmustafa Amin, Xiangtuan Xiong We consider two inverse source problems for the radially symmetric and axis-symmetric heat conduction equations. The source terms depend on the space variables. By a transformation, the original inverse source problems can be transferred to nonlocal boundary problems with standard radially symmetric and axis-symmetric heat equations. Then the fundamental solution methods combined with the Tikhonov regularization method are provided to solve the nonlocal boundary problems. Once the solutions of the nonlocal boundary problems are found then the source terms can be recovered by numerical differentiation. Many numerical examples are shown to illustrate the performance of the proposed methods.

Abstract: Publication date: Available online 2 January 2019Source: Applied Numerical MathematicsAuthor(s): Yunhui Yin, Peng Zhu In this paper, the streamline-diffusion finite element method is applied to a two-dimensional convection-diffuse problem posed on the unit square, using a graded mesh of O(N2) points based on standard Lagrange polynomials of degree k≥1. We prove the method is convergent almost uniformly in the perturbation parameter ϵ, and obtains convergence order O(N−klogk+1(1ϵ)) in a streamline-diffusion norm under certain assumptions. Numerical experiments support the theoretical results.

Abstract: Publication date: Available online 22 December 2018Source: Applied Numerical MathematicsAuthor(s): Feng Liao, Luming Zhang, Tingchun Wang In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger-Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O(τ2+h4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis.

Abstract: Publication date: Available online 21 December 2018Source: Applied Numerical MathematicsAuthor(s): Tianliang Hou, Haitao Leng In this paper, we present a two-grid mixed finite element scheme for distributed optimal control problems governed by stationary Stokes equations. In order to avoid the difficulty caused by the symmetry constraint of the stress tensor, we use pseudostress to replace it. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We first prove that the difference between the interpolation and the numerical solution has superconvergence property for the control u with order h2. Then, using the postprocessing technique, we derive a second-order superconvergent result for the control u. Next, we construct a two-grid mixed finite element scheme and derive a priori error estimates. Finally, a numerical experiment is presented to verify the theoretical results.

Abstract: Publication date: Available online 17 December 2018Source: Applied Numerical MathematicsAuthor(s): Yu Zhang, Hai Bi, Yidu Yang In this paper, for biharmonic eigenvalue problems with clamped boundary condition in Rn which include plate vibration problem and plate buckling problem, we primarily study the two-grid discretization based on the shifted-inverse iteration of Ciarlet-Raviart mixed method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh πh can be reduced to the solution of an eigenvalue problem on a coarser mesh πH and the solution of a linear algebraic system on the fine mesh πh. With a new argument which is not covered by existing work, we prove that the resulting solution still maintains an asymptotically optimal accuracy when H>h≥O(H2). The surprising numerical results show the efficiency of our scheme.

Abstract: Publication date: Available online 6 December 2018Source: Applied Numerical MathematicsAuthor(s): L. Zmour, A. Bouidi In this paper, we consider a numerical method for the White–Metzner model of viscoelastic fluid flows by a combination of the weighted least-squares (WLS) method and the streamline upwind/Petrov–Galerkin (SUPG) method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes-like problem and a linearized constitutive equation using WLS and SUPG, respectively. The elastic viscous split stress formulation (EVSS) introduced in [1] is used for the discretization of the constitutive equation. An a priori error estimate for the WLS/SUPG method is derived and numerical results supporting the estimate are presented. We encounter no limit of the Weissenberg number for all values of ε satisfying 0

Abstract: Publication date: Available online 5 December 2018Source: Applied Numerical MathematicsAuthor(s): Florian Monteghetti, Denis Matignon, Estelle Piot This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss-Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delay differential equation, that quadrature-based discretization methods are spectrally correct, i.e. that they yield an unpolluted and convergent approximation of the essential spectrum linked to the fractional derivative, by contrast with optimization-based methods that can yield polluted spectra whose convergence is difficult to assess.

Abstract: Publication date: Available online 5 December 2018Source: Applied Numerical MathematicsAuthor(s): A. Ludu, A. Raghavendra Stable rotating polygonal hollow patterns were predicted theoretically and measured for Leidenfrost drops confined by circular boundary. Liquid free surface modes occur with different number of edges. This two-dimensional incompressible and inviscid flow is controlled by surface tension, evaporation reaction and gravitation. The model is based on shallow water theory and predicts the existence of sharp rotational polygonal waves as solitary waves solutions. These patterns may represent a signature of universality in shallow water rotating models since similar rotating hollow polygons are noticed in tropical cyclones, fast rotated liquids, Saturn's hexagon, and some plasma systems.

Abstract: Publication date: Available online 30 November 2018Source: Applied Numerical MathematicsAuthor(s): Sergey Charnyi, Timo Heister, Maxim A. Olshanskii, Leo G. Rebholz We study discretizations of the incompressible Navier-Stokes equations, written in the newly developed energy-momentum-angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves energy only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier-Stokes equations in primitive variables.

Abstract: Publication date: Available online 22 November 2018Source: Applied Numerical MathematicsAuthor(s): A. Calini, C.M. Schober, M. Strawn We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schrödinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov–Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.

Abstract: Publication date: Available online 22 November 2018Source: Applied Numerical MathematicsAuthor(s): G.M. Coclite, A. Fanizzi, L. Lopez, F. Maddalena, S.F. Pellegrino In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results.

Abstract: Publication date: Available online 29 October 2018Source: Applied Numerical MathematicsAuthor(s): Corrado Lattanzio, Corrado Mascia, Ramón G. Plaza, Chiara Simeoni This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen–Cahn equation –given byτ∂ttu+(1−τf′(u))∂tu−μ∂xxu=f(u), where τ,μ>0 and f is a cubic-like function with three zeros– represents a notable prototype.Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process, given by the substitution of the Fourier's law with the Maxwell–Cattaneo's law. A corresponding pseudo-kinetic scheme is also proposed for the Guyer–Krumhansl's law.For the Maxwell–Cattaneo case, numerical experiments1 are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves under a sign condition on the damping term g:=1−τf′. Moreover, important evidence of the validity of those results beyond the formal hypotheses g(u)>0 for any u is numerically established.

Abstract: Publication date: Available online 27 October 2018Source: Applied Numerical MathematicsAuthor(s): Deniz Bilman, Thomas Trogdon We compare the performance of well-known numerical time-stepping methods that are widely used to compute solutions of the doubly-infinite Fermi-Pasta-Ulam-Tsingou (FPUT) lattice equations. The methods are benchmarked according to (1) their accuracy in capturing the soliton peaks and (2) in capturing highly-oscillatory parts of the solutions of the Toda lattice resulting from a variety of initial data. The numerical inverse scattering transform method is used to compute a reference solution with high accuracy. We find that benchmarking a numerical method on pure-soliton initial data can lead one to overestimate the accuracy of the method.

Abstract: Publication date: Available online 15 October 2018Source: Applied Numerical MathematicsAuthor(s): Sandra Carillo Third order nonlinear evolution equations, that is the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV) equation and other ones are considered: they all are connected via Bäcklund transformations. These links can be depicted in a wide Bäcklund Chart which further extends the previous one constructed in [22]. In particular, the Bäcklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Bäcklund Chart. That is, such a Bäcklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30] is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Bäcklund chart.

Abstract: Publication date: Available online 2 October 2018Source: Applied Numerical MathematicsAuthor(s): Evgueni Dinvay, Denys Dutykh, Henrik Kalisch In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.In the current work, we review some of the existing fully dispersive systems, such as found in [1], [3], [8], [17], [23], [24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

Abstract: Publication date: Available online 2 October 2018Source: Applied Numerical MathematicsAuthor(s): A. Alexandrou Himonas, Dionyssios Mantzavinos, Fangchi Yan The initial-boundary value problem on the half-line for the nonlinear Schrödinger equation supplemented with a Neumann boundary condition is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces Hs. The proof takes advantage of a novel explicit solution formula for the forced linear counterpart of the nonlinear problem, which is obtained via Fokas' unified transform method. This formula provides the basis for setting up a Picard iteration scheme and for deriving the linear estimates required for proving local well-posedness of the nonlinear problem via a contraction mapping argument.

Abstract: Publication date: Available online 24 July 2018Source: Applied Numerical MathematicsAuthor(s): Iftikhar Ahmad, Hira Ilyas In this paper, Homotopy Perturbation Method is applied to solve the nonlinear MHD Jeffery–Hamel arterial blood flow problem. Primarily, two-dimensional nonlinear Navier–Stokes equations have been converted into third order one-dimensional equation by means of transformation rule. Later the solution of governed equation is obtained by using Homotopy Perturbation Method. The proposed numerical results show a good agreement with reference solution for finite interval and emphasize to understand the human arterial blood flow rate. Further, accuracy and reliability of the proposed method is checked by increasing the iteration process up to third order. Finally, the results showed that product of angle between plates “α” and Reynolds number “Re” is directly proportional to the MHD Jeffery–Hamel flow.

Abstract: Publication date: Available online 23 July 2018Source: Applied Numerical MathematicsAuthor(s): Shenghao Li, Min Chen We prove the existence of a large family of two-dimensional diamond shaped standing waves for a Boussinesq system which describes two-way propagation of water waves in a channel. Our proof uses the Lyapunov-Schmidt method to find the bifurcation standing waves.

Abstract: Publication date: Available online 27 June 2018Source: Applied Numerical MathematicsAuthor(s): Runzhang Xu, Wei Lian, Xiangkun Kong, Yanbing Yang The main goal of this work is to investigate the initial boundary value problem of fourth order wave equation with nonlinear strain and logarithmic nonlinearity at three different initial energy levels, i.e., subcritical energy E(0)d. First, we prove the local existence of weak solution by Galerkin method. In the framework of potential well, we obtain the global existence and infinite time blow up of the solution with sub-critical initial energy. Moreover by the scaling technique, we obtain global existence and infinite time blow up of the solution with critical initial energy. Also, a high energy infinite time blow up result is established.

Abstract: Publication date: Available online 6 June 2018Source: Applied Numerical MathematicsAuthor(s): Yongbing Luo, Yanbing Yang, Md Salik Ahmed, Tao Yu, Mingyou Zhang, Ligang Wang, Huichao Xu This paper investigates the local existence, global existence and finite time blow up of the solution to the Cauchy problem for a class of nonlinear Klein–Gordon equation with general power-type nonlinearities. We give some sufficient conditions on the initial data such that the solution exists globally or blows up in finite time with low initial energy and critical energy. Further a finite time blow up result of the solution with high initial energy is proved.

Abstract: Publication date: Available online 30 May 2018Source: Applied Numerical MathematicsAuthor(s): V.S. Gerdjikov, M.D. Todorov We analyze the dynamical behavior of the N-soliton train in the adiabatic approximation of the perturbed nonlinear Schrödinger equation (NLSE) and the Manakov model. The perturbations include the simultaneous by a periodic external potential, and linear and nonlinear gain/loss terms. We derive the corresponding perturbed complex Toda chain (PCTC) models for both NLSE and Manakov model. We show that the soliton interactions dynamics for the PCTC models compares favorably to full numerical results of the original perturbed NLSE and Manakov model.

Abstract: Publication date: Available online 18 May 2018Source: Applied Numerical MathematicsAuthor(s): Bin Liu, Lu Li, Boris A. Malomed A class of periodic solutions of the nonlinear Schrödinger equation with non-Hermitian potentials are considered. The system may be implemented in planar nonlinear optical waveguides carrying an appropriate distribution of local gain and loss, in a combination with a photonic-crystal structure. The complex potential is built as a solution of the inverse problem, which predicts the potential supporting required periodic solutions. The main subject of the analysis is the spectral structure of the linear (in)stability for the stationary spatially periodic states in the periodic potentials. The stability and instability bands are calculated by means of the plane-wave-expansion method, and verified in direct simulations of the perturbed evolution. The results show that the periodic solutions may be stable against perturbations in specific Floquet–Bloch bands, even if they are unstable against small random perturbations.

Abstract: Publication date: Available online 17 May 2018Source: Applied Numerical MathematicsAuthor(s): P.N. Davis, P. van Heijster, R. Marangell We investigate the point spectrum associated with travelling wave solutions in a Keller–Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise.

Abstract: Publication date: Available online 11 April 2018Source: Applied Numerical MathematicsAuthor(s): Sunao Murashige This paper describes a new type of long wave model for periodic internal waves propagating in permanent form at the interface between two immiscible inviscid fluids. This model for irrotational plane motion of these waves is derived in the complex velocity potential planes where the flow domains are conformally mapped. Since no smallness assumption of wave amplitude is made and the wave elevation at the interface is represented by a single-valued function of the velocity potential, this model is applicable to large-amplitude motions of which wave profile may overhang. Numerical examples demonstrate that the proposed model can produce overhanging solutions, and variations of solutions with wavelength or wave amplitude are qualitatively similar to those of the full Euler system. It is also pointed out that the kinematic condition at the interface is exactly satisfied in the proposed model for all wave amplitudes, but not in an existing long wave model derived in the physical plane.