Abstract: Publication date: Available online 14 October 2019Source: Applied Numerical MathematicsAuthor(s): Stefano Giani, Mohammed Seaid In this paper we present a multi-hp adaptive discontinuous Galerkin method for 3D simplified PN approximations of radiative transfer in non-gray media capable of reaching accuracies superior to most of methods in the literature. The simplified PN models are a set of differential equations derived based on asymptotic expansions for the integro-differential radiative transfer equation. In a non-gray media the optical spectrum is divided into a finite set of bands with constant absorption coefficients and the simplified PN approximations are solved for each band in the spectrum. At high temperature, boundary layers with different magnitudes occur for each wavelength in the spectrum and developing a numerical solver to accurately capture them is challenging for the conventional finite element methods. Here we propose a class of high-order adaptive discontinuous Galerkin methods using space error estimators. The proposed method is able to solve problems where 3D meshes contain finite elements of different kind with the number of equations and polynomial orders of approximation varying locally on the finite element edges, faces, and interiors. The proposed method has also the potential to perform both isotropic and anisotropic adaptation for each band in the optical spectrum. Several numerical results are presented to illustrate the performance of the proposed method for 3D radiative simulations. The computed results confirm its capability to solve 3D simplified PN approximations of radiative transfer in non-gray media.

Abstract: Publication date: Available online 11 October 2019Source: Applied Numerical MathematicsAuthor(s): Chengjian Zhang, Cui Li, Jingyao Jiang This paper is concerned with numerical analysis and computation for neutral equations with piecewise constant argument (NEPCA). The block boundary value methods (BBVMs) are extended to solve NEPCA. Under the conditions of preconsistency and p-order consistency, it is proved that the convergence order of the extended BBVMs can arrive at p. Moreover, a sufficient and necessary condition of the numerical asymptotical stability is derived for linear NEPCA. The presented numerical examples further testify the computational effectiveness of the methods and the obtained theoretical results.

Abstract: Publication date: Available online 10 October 2019Source: Applied Numerical MathematicsAuthor(s): Xiaoqin Shen, Qian Yang, Linjin Li, Zhiming Gao, Tiantian Wang In this paper, the dynamic Koiter's model is discussed for the hyperbolic parabolic shell, i.e., the saddle surface as its middle surface. The numerical computation for this kind of shell is the most difficult as its complex differential geometry. We provide a numerical algorithm for solving the dynamic problems with finite elements and finite difference methods. At the same time, we do the numerical experiments for the hyperbolic parabolic shell. Concretly, we consider the roof of the Valencia Aquarium as a hyperbolic parabolic shell with a geometric model of the roof deduced. Then, we simulate the deformation of the roof by the dynamic applied force coupled with the dynamic Koiter's model. Finally, a comparison of the displacement distribution on the middle surface is made, which verifies that the dynamic Koiter's model is efficient and that numerical algorithm is stable for the hyperbolic parabolic shell. This result provides a strong theoretical and applicable support for the design and protection of such large-span buildings whose roofs and walls are of one whole object.

Abstract: Publication date: Available online 10 October 2019Source: Applied Numerical MathematicsAuthor(s): Harendra Singh, D. Baleanu, H.M. Srivastava, Hemen Dutta, Navin Kumar Jha In this article, we construct approximate solution to multi-dimensional Fredholm integral equations of second kind using n-dimensional Legendre scaling functions. Error analysis of the problem is provided in the L2 sense. It is shown that our numerical method is numerically stable. Some examples are discussed based on proposed method to show the importance and accuracy of the proposed numerical method.

Abstract: Publication date: Available online 9 October 2019Source: Applied Numerical MathematicsAuthor(s): Yayu Guo, Hongen Jia, Jichun Li, Ming Li In this paper, a fully discrete scheme is proposed for solving the Cahn-Hilliard-Hele-Shaw system with a logarithmic potential and concentration dependent mobility. Using the regularization procedure, the domain for the logarithmic free energy density function F(ϕ) is extended from (−1,1) to (−∞,∞). A convex-splitting method of the energy is adopted in time and the mixed finite element method is used in space. Furthermore, we prove the stability of the proposed numerical method and establish the optimal error estimate in the energy norm. Finally, numerical results are presented to support our theoretical analysis.

Abstract: Publication date: Available online 8 October 2019Source: Applied Numerical MathematicsAuthor(s): Davoud Mirzaei In this paper the error analysis of the kernel collocation method for partial differential equations on the unit sphere is presented. A simple analysis is given when the true solutions lie in arbitrary Sobolev spaces. This also extends the previous studies for true solutions outside the associated native spaces. Finally, some experimental results support the theoretical error bounds.

Abstract: Publication date: Available online 8 October 2019Source: Applied Numerical MathematicsAuthor(s): Maryam Kamranian, Mehdi Tatari, Mehdi Dehghan This paper introduces an element free Galerkin method for solving Stokes equations based on the velocity-pressure formulation. A consistently stabilized element free Galerkin method is proposed and Nitsche's method is applied. We formulate and study the optimal convergence of this method. For this purpose, the concept of the vector-valued moving least squares approximation is derived to derive the error estimation in Sobolev spaces. Finally we report some numerical experiments to illustrate the accuracy of the proposed method.

Abstract: Publication date: Available online 4 October 2019Source: Applied Numerical MathematicsAuthor(s): Haixia Dong, Wenjun Ying, Jiwei Zhang This paper presents a rigorous convergence analysis in ℓ2-norm and maximum norm for the Marker-And-Cell (MAC) scheme for solving the incompressible Stokes equations with Dirichlet boundary conditions in two dimensions. The numerical scheme is based on a finite-difference approximation on a staggered grid, where the computational boundary conditions are introduced by reflecting boundary conditions. This scheme is well known to numerically produce second order accuracy for velocity, pressure as well as the gradient of velocity. However, the reflecting boundary conditions lead to the fact that the truncation errors near the boundary is the order of O(1), which presents great difficulties in error analysis to obtain optimal order. In this work, by means of a discrete LBB condition and some auxiliary functions that satisfy discrete Stokes equations to a high-order accuracy, we first prove that the convergence is second order in the ℓ2-norm for velocity, pressure and the gradient of the velocity. After that, according to the property of discrete Green functions, we further provide rigorous analysis to show second-order accuracy in the maximum norm for both the velocity and its gradient. The extension of our results to three dimensional problems would be routine. Numerical results are also provided to demonstrate our theoretical analysis.

Abstract: Publication date: Available online 3 October 2019Source: Applied Numerical MathematicsAuthor(s): F. Shayanfard, H. Laeli Dastjerdi, F.M. Maalek Ghaini In this paper, we have adapted the collocation method for solving a special class of Volterra integro-differential equations of the third-kind. Structure of the method and its applicability for proposed equations have been presented. Moreover we have investigated the solvability and convergence analysis of the method. Also some numerical examples are provided to illustrate the theoretical results.

Abstract: Publication date: Available online 3 October 2019Source: Applied Numerical MathematicsAuthor(s): Donatella Occorsio, Maria Grazia Russo The paper deals with the numerical approximation of integrals of the typeI(f,y):=∫−11f(x)k(x,y)dx,y∈S⊂R where f is a smooth function and the kernel k(x,y) involves some kinds of “pathologies” (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a “mixed ” quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

Abstract: Publication date: Available online 8 July 2019Source: Applied Numerical MathematicsAuthor(s): Luisa Fermo, Cornelis van der Mee, Sebastiano Seatzu The authors propose a numerical method to approximate the solution of specific bivariate Volterra integral equations which arise in the numerical solution of the initial value problem for the Korteweg-de Vries equation. A preliminary study of the domain of the unknown function is carried out and a general algorithm based on the Gauss-Legendre and Newton-Cotes quadrature formulae is developed. Numerical tests are also given in order to show the efficiency of the method.

Abstract: Publication date: Available online 27 June 2019Source: Applied Numerical MathematicsAuthor(s): Mikhail Bulatov, Liubov Solovarova We consider the initial value problem in linear differential-algebraic equations. We propose collocation-variation difference schemes with several collocation points and show the principal difference between these approaches and the known difference schemes. Analysis of the particular cases of such schemes and calculations of the test examples are given.

Abstract: Publication date: Available online 18 June 2019Source: Applied Numerical MathematicsAuthor(s): F. Iavernaro, F. Mazzia, M.S. Mukhametzhanov, Ya.D. Sergeyev Multi-derivative one-step methods based upon Euler–Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge–Kutta methods, we show that the Euler–MacLaurin method of order p is conjugate-symplectic up to order p+2. This feature entitles them to play a role in the context of geometric integration and, to make their implementation competitive with the existing integrators, we explore the possibility of computing the underlying higher order derivatives with the aid of the Infinity Computer.

Abstract: Publication date: Available online 14 June 2019Source: Applied Numerical MathematicsAuthor(s): Giuseppe Mastroianni, Incoronata Notarangelo The paper deals with weighted polynomial approximation for functions defined on (−1,1), which can grow exponentially both at −1 and at 1. We summarize recent results on function spaces with new moduli of smoothness, estimates for the best approximation, Lagrange interpolation, Fourier sums and Gaussian rules with respect to weights of the form w(x)=(1−x2)βe−(1−x2)−α.

Abstract: Publication date: Available online 31 May 2019Source: Applied Numerical MathematicsAuthor(s): João C. Matos, José A. Matos, Maria João Rodrigues, Paulo B. Vasconcelos Computing approximate solutions of integro-differential problems can be difficult and time consuming, requiring large expertize. Tau Toolbox is a numerical library that produces approximate polynomial solutions of differential, integral and integro-differential equations via the spectral Lanczos' Tau method. This approach, however, may fail to ensure the spectral rate of convergence, and even to reach convergence, whenever the exact solution shows singularities. This paper describes a post-processing phase based on a Frobenius-Padé approximation method to build rational approximations from the polynomial Tau approximation. This filtering extension improves the accuracy of the spectral approximation when working in the vicinity of solutions with singularities. Furthermore, some insights on how to perform such a filtering process on a piecewise version of the Tau method to tackle larger domains and/or stiff problems are offered.

Abstract: Publication date: Available online 30 May 2019Source: Applied Numerical MathematicsAuthor(s): S. Nemati, P.M. Lima, S. Sedaghat In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss–Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss–Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.

Abstract: Publication date: Available online 23 May 2019Source: Applied Numerical MathematicsAuthor(s): Eleonora Messina, Eugenio Chioccarelli, Georgios Baltzopoulos, Antonia Vecchio The dynamic behaviour of rigid blocks subjected to support excitation is represented by discontinuous differential equations with state jumps. In the numerical simulation of these systems, the jump times corresponding to the numerical trajectory do not coincide with the ones of the given problem. When multiple state jumps occur, this approximation may affect the accuracy of the solution and even cause an order reduction in the method. Focus here is on the error behaviour in the numerical dynamic. The basic idea is to investigate how the error propagates in successive impacts by decomposing the numerical integration process of the overall system into a sequence of discretized perturbed problems.

Abstract: Publication date: Available online 7 May 2019Source: Applied Numerical MathematicsAuthor(s): P. Junghanns, R. Kaiser We discuss the problem of the instability of the numerical method proposed by Kalandiya [10] for the approximate solution of the two-dimensional elasticity problem of a crack at a circular cavity surface. Furthermore, we describe a method to overcome this problem.

Abstract: Publication date: Available online 26 April 2019Source: Applied Numerical MathematicsAuthor(s): E.V. Chistyakova, V.F. Chistyakov This paper addresses systems of linear integral differential equations with a singular matrix multiplying the higher derivative of the desired vector-function. Such systems can be viewed as differential algebraic equations perturbed by the Fredholm operators. For such problems, we obtain solvability conditions and discuss the influence of small perturbations of the input data on the solution. Within the existence theorems, we propose the least squares method to obtain a numerical approximation.

Abstract: Publication date: Available online 24 April 2019Source: Applied Numerical MathematicsAuthor(s): Christopher T.H. Baker, Neville J. Ford The material here is motivated by the discussion of solutions of linear homogeneous and autonomous differential equations with deviating arguments. If a,b,c and {τˇℓ} are real and γ♮ is real-valued and continuous, an example with these parameters is(⋆)u′(t)={au(t)+bu(t+τˇ1)+cu(t+τˇ2)}+∫τˇ3τˇ4γ♮(s)u(t+s)ds. A wide class of equations (⋆), or of similar type, can be written in the “canonical” form(⋆⋆)u′(t)=∫τminτmaxu(t+s)dσ(s)(t∈R), for a suitable choice of τmin,τmax where σ is of bounded variation and the integral is a Riemann-Stieltjes integral. For equations written in the form (⋆⋆), there is a corresponding characteristic function(⋆⋆⋆)χ(ζ)):=ζ−∫τminτmaxexp(ζs)dσ(s)(ζ∈

Abstract: Publication date: Available online 12 April 2019Source: Applied Numerical MathematicsAuthor(s): Lorenzo Mascotto, Alexander Pichler We extend the nonconforming Trefftz virtual element method introduced in [30] to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers; secondly, we enrich such local spaces with special functions capturing the physical behavior of the solution to the target problem. As these two issues are directly related to an increase of the number of degrees of freedom, we use a reduction strategy inspired by [31], which allows to mitigate the growth of the dimension of the approximation space when considering h- and p-refinements. This renders the new method highly competitive in comparison to other Trefftz and quasi-Trefftz technologies tailored for the Helmholtz problem with piecewise constant wave number. A wide range of numerical experiments, including the p-version with quasi-uniform meshes and the hp-version with isotropic and anisotropic mesh refinements, is presented.

Abstract: Publication date: Available online 3 April 2019Source: Applied Numerical MathematicsAuthor(s): F. Dassi, G. Vacca We present the essential tools to deal with virtual element method (VEM) for the approximation of solutions of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. We exploit such “bricks” to construct virtual element approximations of Stokes, Darcy and Navier–Stokes problems and we provide a series of examples to numerically verify the theoretical behaviour of high-order VEM.

Abstract: Publication date: Available online 2 October 2019Source: Applied Numerical MathematicsAuthor(s): L. Teng, Aleksandr Lapitckii, Michael Günther A multi-step scheme on time-space grids for solving backward stochastic differential equations was proposed in a previous paper, where Lagrange interpolating polynomials are used to approximate the time-integrands with given values of these integrands at chosen multiple time levels. However, the number of time levels is limited, as the stability is lost if the number of time levels is large. For a better stability and the admission of more time levels, in this work, the application of spline replacing Lagrange interpolating polynomials is investigated to approximate the time-integrands. The resulting scheme is a semi-discretization in the time direction involving conditional expectations, which can be numerically solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for arbitrarily many multiple time levels without loosing stability. Several numerical examples including applications in finance are presented to show better approximation properties of this adapted scheme.

Abstract: Publication date: Available online 27 September 2019Source: Applied Numerical MathematicsAuthor(s): A. Chakib, A. Hadri, A. Nachaoui, M. Nachaoui This paper deals with multiscale analysis, by using the correctors, of a nonlinear heat transmission problem in a periodic microscopic structure with multiple components. The occurring nonlinearity is related to the flux condition at the interface between the two parts of the medium. In this work, we present first the multiscale asymptotic expansion of the solution for this kind of problems and the strong convergence of the asymptotic expansion to the solution of the multiscale model. Then, we introduce a numerical algorithm based on the multiscale method for solving this problem. Finally, in order to confirm the efficiency of the proposed algorithm, some numerical results obtained through finite element approximations are presented.

Abstract: Publication date: Available online 25 September 2019Source: Applied Numerical MathematicsAuthor(s): Sanghyun Lee, Woocheol Choi We present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and L2 norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results.

Abstract: Publication date: Available online 23 September 2019Source: Applied Numerical MathematicsAuthor(s): Yunying Zheng, Zhengang Zhao In this paper a time discontinuous space-time finite element method for fractional diffusion-wave equation is studied. The existence and the uniqueness of the numerical solution are included. When we adopt the shape function as the piecewise r-order function, the convergence order is proved to be r+1 in the sense of L2 norm and r+1/2 in the sense of max norm, respectively. Numerical examples with comparisons the accuracy and effectiveness of proposed method are presented.

Abstract: Publication date: Available online 20 September 2019Source: Applied Numerical MathematicsAuthor(s): Dongyang Shi, Ran Wang This paper aims to present a two-grid method (TGM) with low order nonconforming EQ1rot finite element for solving a class of nonlinear wave equations, and to give the superconvergent error analysis unconditionally. Firstly, for the Crank-Nicolson fully discrete scheme, the existence and uniqueness of the numerical solutions are proved, and based on the special characters of this element as well as the priori estimates, the supercloseness and superconvergence analysis of both the original variable u and the auxiliary variable q=ut in the broken H1-norm are deduced on the coarse mesh for the Galerkin finite element method (FEM). Then, by employing the interpolation postprocessing approach and the boundness of the numerical solution in the broken H1-norm on the coarse mesh, the corresponding superconvergence results of order O(τ2+H4+h2) for the TGM are obtained unconditionally, here τ, H and h denote time step, coarse and fine grid sizes, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

Abstract: Publication date: Available online 19 September 2019Source: Applied Numerical MathematicsAuthor(s): Sabrina Francesca Pellegrino In this paper we validate the implementation of the numerical scheme proposed in [3]. The validation is made by comparison with an explicit solution here obtained, and the solutions of Riemann problems for several networks. We then perform some simulations in order to qualitatively validate the model under consideration.Such results represent also a first step for the validation of the finite volumes scheme introduced in [14].

Abstract: Publication date: Available online 17 September 2019Source: Applied Numerical MathematicsAuthor(s): Francesco Dell'Accio, Filomena Di Tommaso The problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular Shepard method, we introduce an hexagonal Shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit.

Abstract: Publication date: Available online 17 September 2019Source: Applied Numerical MathematicsAuthor(s): Quanwei Ren, Hongjiong Tian In this paper, we are concerned with numerical methods for solving stochastic differential equations with Poisson-driven jumps. We construct a class of compensated θ-Milstein methods and study their mean-square convergence and asymptotic mean-square stability. Sufficient and necessary conditions for the asymptotic mean-square stability of the compensated θ-Milstein methods when applied to a scalar linear test equation are derived. We compare the asymptotic mean-square stability region of the linear test equation with that of the compensated θ-Milstein methods with different θ values. Numerical results are given to verify our theoretical results.

Abstract: Publication date: Available online 16 September 2019Source: Applied Numerical MathematicsAuthor(s): J.A. Roberts, N.I. Kavallaris, A.P. Rowntree We perform a stability analysis on a discrete analogue of a known, continuous model of mutualism. We illustrate how the introduction of delays affects the asymptotic stability of the system's positive nontrivial equilibrium point. In the second part of the paper we explore the insights that the model can provide when it is used in relation to interacting financial markets. We also note the limitations of such an approach.

Abstract: Publication date: Available online 12 September 2019Source: Applied Numerical MathematicsAuthor(s): Mostafa Abbaszadeh, Mehdi Dehghan The main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the temporal direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results.

Abstract: Publication date: Available online 11 September 2019Source: Applied Numerical MathematicsAuthor(s): M.V. Kutniv, B.Y. Datsko, A.V. Kunynets, A. Włoch A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [0,a] with a singularity of the first kind in the point x=0 is proposed. Using the substitution of the independent variable x=et, we reduce the original initial value problem to the one on the interval (−∞,lna]. On some finite irregular grid {tn∈(−∞,lna], n=0,1,...,N,tN=lna} Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t0, new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown.

Abstract: Publication date: Available online 10 September 2019Source: Applied Numerical MathematicsAuthor(s): Ross Glandon, Paul Tranquilli, Adrian Sandu Many scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short two-term recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.

Abstract: Publication date: Available online 6 September 2019Source: Applied Numerical MathematicsAuthor(s): Nicolas Gillis, Michael Karow, Punit Sharma Consider a discrete-time linear time-invariant descriptor system Ex(k+1)=Ax(k) for k∈Z+. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system Eˆx(k+1)=Aˆx(k) with rank(Eˆ)=r. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.

Abstract: Publication date: Available online 5 September 2019Source: Applied Numerical MathematicsAuthor(s): Afaf Bouharguane, Nour Seloula In this paper, we consider the discontinuous Galerkin method for solving time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order α∈(1,2). These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering model. The key to study these numerical schemes is to split the anti-diffusive operators into a singular and non-singular integral representations. The problem is then expressed as a system of low order differential equations and a local discontinuous Galerkin method is proposed for these equations. We prove nonlinear stability estimates and optimal order of convergence O(Δxk+1) for linear equations and an order of convergence of O(Δxk+12) for the nonlinear problem. Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme our convergence results.

Abstract: Publication date: Available online 4 September 2019Source: Applied Numerical MathematicsAuthor(s): Md. Maqbul, A. Raheem In this paper, we use the Rothe time-discretization method to prove the existence and uniqueness of a strong solution of a semilinear pseudo-parabolic equation with integral conditions. We also provided an example to illustrate the main result.

Abstract: Publication date: Available online 4 September 2019Source: Applied Numerical MathematicsAuthor(s): Deepika Sharma, Kavita Goyal, Rohit Kumar Singla This paper proposes a fast curvelet based finite difference method for numerical solutions of partial differential equations (PDEs). The method uses finite difference approximations for differential operators involved in the PDEs. After the approximation, the curvelet is used for the compression of the finite difference matrices and subsequently for computing the dyadic powers of these matrices required for solving the PDE in a fast and efficient manner. As a prerequisite, compression and reconstruction errors for the curvelet have been tested against different parameters. The developed method has been applied on five test problems of different nature. For each test problem the convergence of the method is examined. Moreover, to measure the performance of the proposed method the computational time taken by the proposed method is compared to that of the finite difference method. It is observed that the proposed method is computationally very efficient.

Abstract: Publication date: Available online 3 September 2019Source: Applied Numerical MathematicsAuthor(s): S.M. Abrarov, B.M. Quin In our previous publications [10] we have introduced the cosine product-to-sum identity∏m=1Mcos(t2m)=12M−1∑m=12M−1cos(2m−12Mt) and applied it for sampling [8], [9] as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only 16 summation terms can provide accuracy ∼10−14. In this work we generalize this approach and show as an example how a rational approximation of the sinc function can be derived. A MATLAB code validating these results is presented.

Abstract: Publication date: Available online 2 September 2019Source: Applied Numerical MathematicsAuthor(s): Pratibhamoy Das, Subrata Rana, Jesus Vigo-Aguiar In the present research, we consider a boundary layer originated system of reaction diffusion problems whose boundary conditions are of mixed type. This problem is singularly perturbed nature and the diffusion parameters are considered to be of different magnitude. We develop two adaptive methods based on r- refinement strategy, which move the mesh points toward the boundary layers. Here, we use a curvature based monitor function for adaptive moving mesh generation. Based on a combination of forward and backward difference schemes on this adaptively generated equidistributed mesh, we obtain a first order uniformly accurate solution. The discrete solution can be enhanced to a second order uniform accuracy by our proposed cubic spline based scheme. Numerically, we provide a comparison of our present method with the commonly used meshes. Experiments show the highly effective behavior of the present method.

Abstract: Publication date: Available online 30 August 2019Source: Applied Numerical MathematicsAuthor(s): Aihua Lin, Anastasiia Kuzmina, Per Kristen Jakobsen In this work, we introduce a method for solving linear and nonlinear scattering problems for wave equations using a new hybrid approach. This new approach consists of a reformulation of the governing equations into a form that can be solved by a combination of a domain-based method and a boundary-integral method. Our reformulation is aimed at a situation in which we have a collection of compact scattering objects located in an otherwise homogeneous unbounded space.The domain-based method is used to propagate the equations governing the wave field inside the scattering objects forward in time. The boundary integral method is used to supply the domain-based method with the required boundary values for the wave field.In this way, the best features of both methods come into play. The response inside the scattering objects, which can be caused by both material inhomogeneity and nonlinearities, is easily considered using the domain-based method, and the boundary conditions supplied by the boundary integral method makes it possible to confine the domain-method to the inside of each scattering object.

Abstract: Publication date: Available online 30 August 2019Source: Applied Numerical MathematicsAuthor(s): Dongyang Shi, Xu Jia In this paper, a new second-order accurate backward differentiation formula of the two-grid finite element method is developed for the nonlinear Benjamin-Bona-Mahony equation, and the stability and unconditional superconvergence analysis of this scheme are presented. This method firstly solves a nonlinear system on a coarse mesh with size H to get the numerical solution UHn and then computes a linear system constructed by the previous result UHn on a fine mesh with size h(h

Abstract: Publication date: Available online 29 August 2019Source: Applied Numerical MathematicsAuthor(s): Matt Majic A new boundary integral approach is proposed for evaluating the potential of an arbitrary given static mass/charge distribution with a definite boundary, reducing the problem from a volume to a surface integral. In this approach, the internal potential is split into a particular solution to Poisson's equation plus some solution to Laplace's equation. The surface integral solution is then derived from Green's identities and the boundary conditions. For rotational symmetry the surface integral may be reduced to a line integral. Using this approach, the gravitational potential of a homogeneous torus is expressed as a series of toroidal harmonics, and results are confirmed with the solution obtained via volume integration. Analytical results are also derived for a torus with a mass distribution concentrated around the plane of symmetry. The surface integral approach is tested numerically against the standard volume integral for a spheroid.

Abstract: Publication date: Available online 28 August 2019Source: Applied Numerical MathematicsAuthor(s): Dongyang Shi, Qian Liu In this paper, unconditional superconvergent estimate with a Galerkin finite element method (FEM) is presented for Ginzburg-Landau equation by conforming bilinear FE, while all previous works require certain time-step restrictions. First of all, a time-discrete system is introduced, with which the error function is split into a temporal error and a spatial error. On one hand, the regularity of the time-discrete system is deduced with a rigorous analysis. On the other hand, the error between the numerical solution and the solution of the time-discrete system is derived τ-independently with order O(h2+hτ), where h is the subdivision parameter and τ, the time step. Then, the unconditional superclose result of order O(h2+τ) in the H1-norm is deduced directly based on the above estimates. Furthermore, the global superconvergent result is obtained through the interpolated postprocessing technique. At last, numerical results are provided to confirm the theoretical analysis.

Abstract: Publication date: Available online 28 August 2019Source: Applied Numerical MathematicsAuthor(s): Zhendong Luo, Wenrui Jiang In this paper, by using proper orthogonal decomposition (POD) to reduce the order of the coefficient vector of the classical Crank–Nicolson finite spectral element (CCNFSE) method for the two-dimensional (2D) non-stationary Navier-Stokes equations about vorticity-stream functions, we first establish a reduced-order extrapolated Crank–Nicolson finite spectral element (ROECNFSE) method for the 2D non-stationary Navier-Stokes equations so that the ROECNFSE method has the same basis functions as the CCNFSE method and maintains all the advantages of the CCNFSE method. Then, by means of matrix analysis, we analyze the existence, stability, and convergence of the ROECNFSE solutions such that the theoretical analysis becomes very simple and convenient. 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 ROECNFSE method are further verified. Therefore, both theory and method of this paper is new and completely different from the existing reduced-order methods.

Abstract: Publication date: Available online 28 August 2019Source: Applied Numerical MathematicsAuthor(s): Theodore C. Holtom, Anthony C. Brooms We consider the matter of how to assess uncertainty propagation for the converging beam triple LIDAR technology, which is used for measuring wind velocity passing through a fixed point in space. Converging beam triple LIDAR employs the use of three non-parallel, non-coplanar, laser beams which are directed from a fixed platform, typically at ground level, that extend to meet at the point at which measurement of velocity is sought. Coordinate values of the velocity are ascertained with respect to unit vectors along the lines of sight of the laser beams (Doppler vectors), which are then resolved in order to determine the velocity in terms of Cartesian coordinates (i.e. with respect to the standard basis). However, if there is any discrepancy between the recorded values of the coordinates with respect to the Doppler unit vectors and/or the perceived angle settings for such vectors with what they really should be, however small, then this will lead to errors in the reconstructed Cartesian coordinates. One aim of this paper is to present the detailed formulae that would be required for the computation of the estimated variances of the reconstructed velocities, calculated on the basis of the error propagation formulae (and to highlight to the practitioner the conditions under which such estimated variances could be relied upon). The other main aim of this paper is to demonstrate that for (various wind profiles and, in particular) certain LIDAR configurations, which can be characterized by an associated parallelepiped with unit edge length, the estimated variances have the potential to be unsuitably large, thus indicating that the reconstructed velocity may be unreliable for gauging the value of the true wind velocity and that the practitioner should avoid such configurations in the measurement campaign.

Abstract: Publication date: Available online 27 August 2019Source: Applied Numerical MathematicsAuthor(s): Kamran Kazmi, Abdul Q.M. Khaliq A split-step predictor-corrector method, based on Adams-Moulton formula, is presented for the solution of nonlinear distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions. The distributed-order space-fractional equation is first discretized to a multi-term space-fractional equation by applying a quadrature rule. Multi-term space-fractional equation is spatially discretized by matrix transfer technique and a linearly implicit split-step predictor-corrector method is applied for time-stepping to avoid solving nonlinear systems at each time step. The method is shown to be unconditionally stable and second order convergent. Numerical experiments are performed to confirm the stability and second order convergence of the method. The split-step predictor-corrector method is also compared with a second order IMEX predictor-corrector method. The IMEX predictor-corrector method is found to incur oscillatory behavior when implemented on problems with nonsmooth initial conditions or with mismatch between the initial and boundary conditions. Our method, which is an L-stable method, produces reliable and oscillation free results for such problems.

Abstract: Publication date: Available online 27 August 2019Source: Applied Numerical MathematicsAuthor(s): Ru-Ru Ma, Zheng-Jian Bai In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method.

Abstract: Publication date: Available online 27 August 2019Source: Applied Numerical MathematicsAuthor(s): G.Yu. Kulikov, M.V. Kulikova This paper further advances the idea of accurate Gaussian filtering towards efficient unscented-type Kalman methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the differential equations evolving sigma points utilized in computations of the predicted mean and covariance in time-propagations of the Gaussian distribution are solved accurately, i.e. with negligible error. The latter allows the total error of the unscented Kalman filtering technique to be reduced significantly and gives rise to the novel accurate continuous-discrete unscented Kalman filtering algorithm. At the same time, this algorithm is rather vulnerable to round-off and numerical integration errors committed in each state estimation run because of the need for the Cholesky decomposition of covariance matrices involved. Such a factorization will always fail when the covariance's positivity is lost. This positivity lost issue is commonly resolved with square-root filtering implementations, which propagate not the full covariance matrix but its square root (Cholesky factor) instead. Unfortunately, negative weights encountered in applications of the accurate continuous-discrete unscented Kalman filter to high-dimensional stochastic systems preclude from designing conventional square-root methods. In this paper, we address this problem with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined numerically in an air traffic control scenario.

Abstract: Publication date: Available online 27 August 2019Source: Applied Numerical MathematicsAuthor(s): Zhiyue Zhang, Dong Liang, Quanxiang Wang In this paper, we develop the immersed finite element method for parabolic optimal control problems with interfaces. By employing the definition of directional derivative of Lagrange function, first-order necessary optimality conditions in qualified form for parabolic optimal control problems with interfaces are established. The parabolic state equations are treated with the immersed finite element method using a simple uniform mesh which is independent of the interface. In the interface elements, functions are locally piecewise bilinear functions according to the subelements formed by the actual interface curves. By introducing the auxiliary functions which are the solutions of interface parabolic equations with non-homogenous and homogeneous jump conditions, optimal error estimates are proved for the proposed schemes to the controls, states and adjoint states in both the semi-discrete case and the fully discrete case. Numerical experiments show the performance of the proposed scheme to solve the parabolic optimal control problems with interfaces and confirm the theoretical results.

Abstract: Publication date: Available online 27 August 2019Source: Applied Numerical MathematicsAuthor(s): Gonglin Yuan, Tingting Li, Wujie Hu Nonlinear systems present a quite complicate problem. As the number of dimensions increases, it becomes more difficult to find the solution of the problem. In this paper, a modified conjugate gradient method is designed that has a sufficient descent property and trust region property. It is interesting that the formula for search direction makes full use of the property of convex combination between the deepest descent algorithm and the classical LS conjugate gradient (CG) method. The global convergence property is established under certain conditions, and the numerical results show that the modified method is effective for normal nonlinear equations and image restoration problems.

Abstract: Publication date: Available online 26 August 2019Source: Applied Numerical MathematicsAuthor(s): Wei Zhang In this paper, we consider the theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The existence, uniqueness and pth moment boundedness of the analytic solutions are investigated. Euler method is shown to be divergent in the strong mean square sense for super linear growth coefficients, so the truncated Euler-Maruyama method is presented and its moment boundedness and Lq-convergence are shown. Moreover, its pth moment boundedness and Lq-convergence (q∈[2,p) and p is a parameter in Khasminskii-type condition) rate are given under Local Lipschitz condition and Khasminskii-type condition. The theoretical results are illustrated by some numerical examples.

Abstract: Publication date: Available online 26 August 2019Source: Applied Numerical MathematicsAuthor(s): Jean Daniel Mukam, Antoine Tambue This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than the autonomous equations when modelling real world phenomena. Such equations find applications in many fields such as transport in porous media, quantum fields theory, electromagnetism and nuclear physics. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case are not yet well understood. Here, a non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the time depending, not necessarily self-adjoint linear operators with the corresponding semi group and provide the strong convergence result of the fully discrete scheme toward the mild solution. The results indicate how the converge orders depend on the regularity of the initial solution and the noise. Additionally, for additive noise we achieve convergence order in time approximately 1 under less regularity assumptions on the nonlinear drift term than required in the current literature, even in the autonomous case. Numerical simulations motivated from realistic porous media flow are provided to illustrate our theoretical finding.

Abstract: Publication date: Available online 23 August 2019Source: Applied Numerical MathematicsAuthor(s): M. Ahmadinia, Z. Safari The aim of the present paper is to introduce a numerical method for time-space fractional sine-Gordon equation. The fractional derivative on the space and on the time are considered in the sense of Reimann–Liouville (of order 1≤β≤2) and in the sense of Caputo (of variable order 1≤α(t)

Abstract: Publication date: Available online 23 August 2019Source: Applied Numerical MathematicsAuthor(s): Sigrun Ortleb A fully discrete L2-stability analysis is carried out for linear advection-diffusion problems in one space dimension, discretized in space by discontinuous Galerkin methods based on the (σ,μ)-family of diffusion schemes and implicit-explicit Runge-Kutta schemes in time. Advection terms are discretized explicitly while diffusion terms are solved implicitly. Conditions on the parameters σ and μ are derived which guarantee L2-stability for time steps Δt=O(d/a2), where a and d denote the advection and diffusion coefficient, respectively, i.e. the allowable time step size does not decrease under grid refinement. These conditions are fulfilled in particular by the BR2 scheme as well as the recent (14,94)-recovery scheme. However, the BR1 scheme and the Baumann-Oden method do not possess a similar stablility property which is also confirmed by numerical experiments.

Abstract: Publication date: Available online 22 August 2019Source: Applied Numerical MathematicsAuthor(s): R. Campagna, S. Cuomo, S. De Marchi, E. Perracchione, G. Severino An elliptic partial differential equation with a singular forcing term, describing a steady state flow determined by a pulse-like extraction at a constant volumetric rate, is approximated by a radial basis function approach which takes advantage of decomposing the original domain. The discretization error of such scheme is numerically estimated and we also face up to instability issues. This produces an effective tool for real applications, as confirmed by comparisons with classical grid-based approaches.

Abstract: Publication date: Available online 21 August 2019Source: Applied Numerical MathematicsAuthor(s): Jiansong Zhang, Xiaomang Shen, Hui Guo, Hongfei Fu, Huiran Han A combined method is provided for simulating the compressible wormhole propagation, in which, the splitting mixed finite element (SMFE) method is applied for the parabolic-type pressure equation, the modified mass-conservative characteristic (MCC) method is presented for the convection-diffusion type solute transport equation, and the traditional finite element method is applied for the porosity equation. The application of the splitting mixed element method results in a symmetric positive definite coefficient matrix of the separated mixed element system and the mass-conservative characteristic finite element method not only does well in handling convection-dominant diffusion problem but also keeps mass balance globally. The convergence of this method is considered and the optimal L2-norm error estimate is also derived. Finally, some numerical examples are provided to confirm theoretical analysis and effectiveness of the proposed method.

Abstract: Publication date: Available online 21 August 2019Source: Applied Numerical MathematicsAuthor(s): Vincenzo Citro, Raffaele D'Ambrosio We analyze long-term properties of stochastic θ-methods for damped linear stochastic oscillators. The presented a-priori analysis of the error in the correlation matrix allows to infer the long-time behaviour of stochastic θ-methods and their capability to reproduce the same long-term features of the continuous dynamics. The theoretical analysis is also supported by a selection of numerical experiments.

Abstract: Publication date: Available online 21 August 2019Source: Applied Numerical MathematicsAuthor(s): Igor Boglaev The paper deals with numerical treatments of systems of nonlinear integro-elliptic equations. Monotone iterative methods, based on the method of upper and lower solutions, are constructed. Convergence of the monotone iterative methods is proved. A construction of initial upper and lower solutions is discussed. Numerical experiments are presented.

Abstract: Publication date: Available online 20 August 2019Source: Applied Numerical MathematicsAuthor(s): Sayer O. Alharbi, Saleh M. Alzahrani The present paper proposes an efficient computational method for solving the time-harmonic multiple scattering problem by many small dielectric circular cylinders. The strategy proposed is based on a reduced integral equation formulation associated to diagonalization properties of the integral operators for the interior problem. In addition, we use an asymptotic approximation of the separation matrices to simplify the formulation. Using a GMRES Krylov iterative solver, accelerated by a preconditioner, we solve in a robust and efficient way the generated linear system. Numerical results support the efficiency and accuracy of the new strategy.

Abstract: Publication date: Available online 20 August 2019Source: Applied Numerical MathematicsAuthor(s): Ke Guo, Deren Han The classical proximal point algorithm (PPA) requires a metric proximal parameter, which is positive definite and symmetric, because it plays the role of the measurement matrix of a norm in the convergence proof. In this paper, our main goal is to show that the metric proximal parameter can be nonsymmetric if the proximal center is shifted appropriately. The resulting nonsymmetric PPA with moving proximal centers maintains the same implementation difficulty and convergence properties as the original PPA, while the nonsymmetry of the metric proximal parameter allows us to design highly customized algorithms that can effectively take advantage of the structures of the model under consideration. We present both the exact and inexact versions of the nonsymmetric PPA with moving proximal centers, and analyze their convergence including the estimate of their worst-case convergence rates measured by the iteration complexity under mild assumptions and their asymptotically linear convergence rates under stronger assumptions.

Abstract: Publication date: Available online 13 August 2019Source: Applied Numerical MathematicsAuthor(s): A. Tamasan, A. Timonov We consider the numerical solution of an inverse conductivity problem that arises in coupled physics conductivity imaging. We propose and develop the method of regularized successive iterations based on a regularized weighted least gradient Robin problem. Unlike the existing numerical techniques, the main novelty is that recovering the electrical conductivity from a single internal data does not require the boundary data. The performance of the proposed method is demonstrated by computer simulations of coupled physics conductivity imaging.

Abstract: Publication date: Available online 8 August 2019Source: Applied Numerical MathematicsAuthor(s): N. Tamang, B. Wongsaijai, T. Mouktonglang, K. Poochinapan In this research, the presented method combines advantages of the finite element and three-level finite difference techniques to obtain the solution of the two-dimensional general Rosenau-RLW equation. An important point is that nevertheless the two-dimensional general Rosenau-RLW is a non-linear equation, but with the proposed scheme we are able to linearize this system and efficiently solve it due to the implicit nature of the system of equations. In addition, the existence and uniqueness of the approximate solution are approved. The convergence and stability of the approximate solution are also examined. The general formulation of the procedure is given and numerical results are carried out to confirm the accuracy of our theoretical results and the efficiency of the scheme.

Abstract: Publication date: Available online 24 July 2019Source: Applied Numerical MathematicsAuthor(s): M. Abundo, G. Ascione, M.F. Carfora, E. Pirozzi We show that the First-Passage-Time probability distribution of a Lévy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show that it preserves the structural properties (non-negativity, boundedness, time monotonicity) of the theoretical solution, having to be a probability distribution. Numerical experiments confirming such findings are reported. Simulations of the sample paths of the considered process are also performed and used to both provide suitable boundary conditions and to validate the numerical results.

Abstract: Publication date: Available online 17 July 2019Source: Applied Numerical MathematicsAuthor(s): Arvet Pedas, Enn Tamme, Mikk Vikerpuur We consider a class of boundary value problems for linear fractional weakly singular integro-differential equations with Caputo fractional derivatives and integral boundary conditions. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution and its Caputo derivative. Based on the obtained regularity properties and by using suitable smoothing transformations along with spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

Abstract: Publication date: Available online 12 July 2019Source: Applied Numerical MathematicsAuthor(s): Luca Dieci In this work we show that numerical integration during sliding motion, for piecewise-smooth systems, can be performed without the need to project the approximate solutions on the constraints' surface. We give a general algorithm, an order of approximation result, and show the effectiveness of the technique on several examples. Comparison between projected and non-projected methods and the standard regularization approach is also given.