Authors:Mahboub Baccouch Pages: 18 - 37 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Mahboub Baccouch In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L 2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the ( p + 1 ) -degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L 2 -norm with order of convergence p + 2 . Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p + 2 in the L 2 -norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for P p polynomials with p ≥ 1 . Several numerical results are presented to validate the theoretical results.

Authors:Andrea Barth; Franz G. Fuchs Pages: 38 - 51 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Andrea Barth, Franz G. Fuchs In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media. Two types of models are presented: The first has a time-dependent coefficient modeled by the Ornstein–Uhlenbeck process. The second has a random field coefficient with a given covariance in space. For the former a formula for the exact solution in terms of moments is derived. In both cases stable numerical schemes are introduced to solve these random partial differential equations. Simulation results including convergence studies conclude the theoretical findings.

Authors:Kazuho Ito Pages: 1 - 20 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Kazuho Ito An energy conserving spectral scheme is presented for approximating the smooth solution of the dynamic elastica with free ends. The spatial discretization of the elastica is done on the basis of Galerkin spectral methods with a Legendre grid. It is established that the scheme has the unique solution and enjoys a spectral accuracy with respect to the size of the spatial grid. Moreover, some results of a numerical simulation are given to verify that the implemented scheme preserves the discrete energy.

Authors:Mariantonia Cotronei; Nada Sissouno Pages: 21 - 34 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Mariantonia Cotronei, Nada Sissouno The aim of the paper is to present Hermite-type multiwavelets, i.e. wavelets acting on vector data representing function values and consecutive derivatives, which satisfy the vanishing moment property with respect to elements in the space spanned by exponentials and polynomials. Such functions satisfy a two-scale relation which is level-dependent as well as the corresponding multiresolution analysis. An important feature of the associated filters is the possibility of factorizing their symbols in terms of the so-called cancellation operator. This is shown, in particular, in the situation where Hermite multiwavelets are obtained by completing interpolatory level-dependent Hermite subdivision operators, reproducing polynomial and exponential data, to biorthogonal systems. A few constructions of families of multiwavelet filters of this kind are proposed.

Authors:Peyman Hessari; Byeong-Chun Shin Pages: 35 - 52 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Peyman Hessari, Byeong-Chun Shin The subject of this paper is to investigate the first order system least squares Legendre and Chebyshev pseudo-spectral methods for coupled Stokes–Darcy equations. By introducing strain tensor as a new variable, Stokes–Darcy equations recast into a system of first order differential equations. The least squares functional is defined by summing up the weighted L 2 -norm of residuals of the first order system for coupled Stokes–Darcy equations. To treat Beavers–Joseph–Saffman interface conditions, the weighted L 2 -norm of these conditions are also added to the least squares functional. Continuous and discrete homogeneous functionals are shown to be equivalent to the combination of weighted H ( div ) and H 1 -norm for Stokes–Darcy equations. The spectral convergence for the Legendre and Chebyshev methods are derived. To demonstrate this analysis, numerical experiments are also presented.

Authors:Yuan-Ming Wang Pages: 53 - 67 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Yuan-Ming Wang This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L 2 , H 1 and L ∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L 2 , H 1 and L ∞ norms.

Authors:Jialin Hong; Lihai Ji; Linghua Kong; Tingchun Wang Pages: 68 - 81 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Jialin Hong, Lihai Ji, Linghua Kong, Tingchun Wang It has been pointed out in literature that the symplectic scheme of a nonlinear Hamiltonian system can not preserve the total energy in the discrete sense Ge and Marsden (1988) [10]. Moreover, due to the difficulty in obtaining a priori estimate of the numerical solution, it is very hard to establish the optimal error bound of the symplectic scheme without any restrictions on the grid ratios. In this paper, we develop and analyze a compact scheme for solving nonlinear Schrödinger equation. We introduce a cut-off technique for proving optimal L ∞ error estimate for the compact scheme. We show that the convergence of the compact scheme is of second order in time and of fourth order in space. Meanwhile, we define a new type of energy functional by using a recursion relationship, and then prove that the compact scheme is mass and energy-conserved, symplectic-conserved, unconditionally stable and can be computed efficiently. Numerical experiments confirm well the theoretical analysis results.

Authors:Xin-He Miao; Jian-Tao Yang; B. Saheya; Jein-Shan Chen Pages: 82 - 96 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Xin-He Miao, Jian-Tao Yang, B. Saheya, Jein-Shan Chen In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

Authors:Hailong Qiu; Rong An; Liquan Mei; Changfeng Xue Pages: 97 - 114 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Hailong Qiu, Rong An, Liquan Mei, Changfeng Xue Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions are considered in this paper. Our algorithms consist of solving one Navier–Stokes variational inequality problem used the linear equal-order finite element pair (i.e., P 1 – P 1 ) and then solving a linearization variational inequality problem used the quadratic equal-order finite element pair (i.e., P 2 – P 2 ). Moreover, the stability and convergence of our two-step algorithms are derived. Finally, numerical tests are presented to check theoretical results.

Authors:Ivan Sofronov Pages: 115 - 124 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Ivan Sofronov In [22] we announced equations for yielding differential operators of transparent boundary conditions (TBCs) for a certain class of second order hyperbolic systems. Here we present the full derivation of these equations and consider ways of their solving. The solutions represent local parts of TBCs, and they can be used as approximate nonreflecting boundary conditions. We give examples of computing such conditions called ‘truncated TBCs’ for 3D elasticity and Biot poroelasticity

Authors:J.A. Ferreira; D. Jordão; L. Pinto Pages: 125 - 140 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): J.A. Ferreira, D. Jordão, L. Pinto In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L 2 -norm.

Authors:Thinh Kieu Pages: 141 - 164 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Thinh Kieu In this paper, we consider the generalized Forchheimer flows for slightly compressible fluids in porous media. Using Muskat's and Ward's general form of Forchheimer equations, we describe the flow of a single-phase fluid in R d , d ≥ 2 by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

Authors:Aleksandr E. Kolesov; Michael V. Klibanov; Loc H. Nguyen; Dinh-Liem Nguyen; Nguyen T. Thành Pages: 176 - 196 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Aleksandr E. Kolesov, Michael V. Klibanov, Loc H. Nguyen, Dinh-Liem Nguyen, Nguyen T. Thành The recently developed globally convergent numerical method for an inverse medium problem with the data resulting from a single measurement, proposed in [23], is tested on experimental data. The data were originally collected in the time domain, whereas the method works in the frequency domain with the multi-frequency data. Due to a significant amount of noise in the measured data, a straightforward application of the Fourier transform to these data does not work. Hence, we develop a heuristic data preprocessing procedure, which is described in the paper. The preprocessed data are used as the input for the inversion algorithm. Numerical results demonstrate a good accuracy of the reconstruction of both refractive indices and locations of targets.

Authors:Hossein Beyrami; Taher Lotfi; Katayoun Mahdiani Pages: 197 - 214 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Hossein Beyrami, Taher Lotfi, Katayoun Mahdiani In this article, we approximate the solution of the weakly singular Volterra integral equation of the second kind using the reproducing kernel Hilbert space (RKHS) method. This method does not require any background mesh and can easily be implemented. Since the solution of the second kind weakly singular Volterra integral equation has unbounded derivative at the left end point of the interval of the integral equation domain, RKHS method has poor convergence rate on the conventional uniform mesh. Consequently, the graded mesh is proposed. Using error analysis, we show the RKHS method has better convergence rate on the graded mesh than the uniform mesh. Numerical examples are given to confirm the error analysis results. Regularization of the solution is an alternative approach to improve the efficiency of the RKHS method. In this regard, an smooth transformation is used to regularization and obtained numerical results are compared with other methods.

Authors:Zhiping Yan; Aiguo Xiao; Xiao Tang Pages: 215 - 232 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Zhiping Yan, Aiguo Xiao, Xiao Tang Neutral stochastic delay differential equations often appear in various fields of science and engineering. The aim of this article is to investigate the strong convergence of the split-step theta (SST) method for the neutral stochastic delay differential equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the SST method with θ ∈ [ 0 , 1 ] strongly converges to the exact solution with the order 1 2 . Some numerical results are presented to confirm the obtained results.

Authors:Helena Zarin Pages: 233 - 242 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Helena Zarin A one-dimensional singularly perturbed boundary value problem with two small perturbation parameters is numerically solved on an exponentially graded mesh. Using an h-version of the standard Galerkin method with higher order polynomials, we prove a robust convergence in the corresponding energy norm. Numerical experiments support theoretical findings.

Authors:Tian-jun Wang; Yu-jian Jiao Pages: 243 - 256 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Tian-jun Wang, Yu-jian Jiao In this paper, a fully discrete pseudospectral scheme for Fisher's equation whose solutions behave differently as x → + ∞ and x → − ∞ is presented using generalized Hermite interpolation. The convergence and the stability of the proposed scheme are analyzed. Numerical results coincide well with theoretical analysis and show efficiency of our algorithm.

Authors:Mário Basto; Teresa Abreu; Viriato Semiao; Francisco L. Calheiros Pages: 257 - 269 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Mário Basto, Teresa Abreu, Viriato Semiao, Francisco L. Calheiros The behavior of an iterative method applied to nonlinear equations may be considerably sensitive to the starting points. Comparisons between iterative methods are supported by the study of the basins of attraction in the complex plane C . However, usually, nothing is said about the rate of convergence. In this paper, by making recourse to several examples of algebraic and transcendental equations, a numerical comparison is performed between three methods with the same structure, namely BSC, Halley's and Euler–Chebyshev's methods. The study takes into account both the basins of attraction and the rate of convergence which is measured as the number of iterations required to obtain an equation root with a given tolerance.

Authors:Cem Çelik; Melda Duman Pages: 270 - 286 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Cem Çelik, Melda Duman A space fractional diffusion equation involving symmetric tempered fractional derivative of order 1 < α < 2 is considered. A Galerkin finite element method is implemented to obtain spatial semi-discrete scheme and first order centered difference in time is used to find a fully discrete scheme for tempered fractional diffusion equation. We construct a variational formulation and show its existence, uniqueness and regularity. Stability and error estimates of numerical scheme are discussed. The theoretical and computational study of accuracy and consistence of the numerical solutions are presented.

Authors:Ajit Patel; Sanjib Kumar Acharya; Amiya Kumar Pani Pages: 287 - 304 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Ajit Patel, Sanjib Kumar Acharya, Amiya Kumar Pani In this paper, we discuss a new stabilized Lagrange multiplier method for finite element solution of multi-domain elliptic and parabolic initial-boundary value problems with non-matching grid across the subdomain interfaces. The proposed method is consistent with the original problem and its stability is established without using the inf-sup (well known as LBB) condition. In the first part of this article, optimal error estimates are derived for second order elliptic interface problems. Then, the analysis is extended to parabolic initial and boundary value problems with interface and optimal error estimates are established for both semi-discrete and completely discrete schemes. The results of numerical experiments support the theoretical results obtained in this article.

Authors:K. Van Bockstal; M. Slodička; F. Gistelinck Pages: 305 - 323 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): K. Van Bockstal, M. Slodička, F. Gistelinck In this contribution, the reconstruction of a solely time-dependent convolution kernel in an nonlinear parabolic equation is studied. The missing kernel is recovered from a global integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. More specific, a numerical algorithm based on Rothe's method is designed. Numerical experiments support the obtained results.

Authors:Haibiao Zheng; Jiaping Yu; Li Shan Pages: 1 - 17 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Haibiao Zheng, Jiaping Yu, Li Shan The unconditional convergence of finite element method for two-dimensional time-dependent viscoelastic flow with an Oldroyd B constitutive equation is given in this paper, while all previous works require certain time-step restrictions. The approximation is stabilized by using the Discontinuous Galerkin (DG) approximation for the constitutive equation. The analysis bases on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element approximation of corresponding iterated time-discrete PDEs. The approach used in this paper can be applied to more general couple nonlinear parabolic and hyperbolic systems.

Authors:Wei Jiang; Na Liu Pages: 18 - 32 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Wei Jiang, Na Liu In this article, we proposed a new numerical method to obtain the approximation solution for the time variable fractional order mobile–immobile advection–dispersion model based on reproducing kernel theory and collocation method. The equation is obtained from the standard advection–dispersion equation (ADE) by adding the Coimbra's variable fractional derivative in time of order γ ( x , t ) ∈ [ 0 , 1 ] . In order to solve this kind of equation, we discuss and derive the ε-approximate solution in the form of series with easily computable terms in the bivariate spline space. At the same time, the stability and convergence of the approximation are investigated. Finally, numerical examples are provided to show the accuracy and effectiveness.

Authors:M. Król; M.V. Kutniv; O.I. Pazdriy Pages: 33 - 50 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): M. Król, M.V. Kutniv, O.I. Pazdriy The three-point difference schemes of high order accuracy for the numerical solving boundary value problems on a semi-infinite interval for systems of second order nonlinear ordinary differential equations with a not self-conjugate operator are constructed and justified. We proved the existence and uniqueness of solutions of the three-point difference schemes and obtained the estimate of their accuracy. The results of numerical experiments which confirm the theoretical results are given.

Authors:Mehdi Dehghan; Mostafa Abbaszadeh Pages: 51 - 66 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Mehdi Dehghan, Mostafa Abbaszadeh In the current manuscript, we consider a fractional partial integro-differential equation that is called fractional evolution equation. The fractional evolution equation is based on the Riemann–Liouville fractional integral. The presented numerical algorithm is based on the following procedures: at first a difference scheme has been used to discrete the temporal direction and secondly the spectral element method is applied to discrete the spatial direction and finally these procedures are combined to obtain a full-discrete scheme. For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound. We use the energy method to analysis the full-discrete scheme. We employ some test problems to show the high accuracy of the proposed technique. Also, we compare the obtained numerical results using the present method with the existing methods in the literature.

Authors:Osman Rasit Isik; Aziz Takhirov; Haibiao Zheng Pages: 67 - 78 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Osman Rasit Isik, Aziz Takhirov, Haibiao Zheng This paper deals with the problem of accelerating convergence to equilibrium for the Navier–Stokes equation using time relaxation models. We show that the BDF2 based semidiscrete solution of the regularized scheme converges to the steady-state solution of the continuous Navier–Stokes equations, under appropriate conditions. The proof also shows that time relaxation model can be used to accelerate the convergence with the appropriate choice of the parameters. Numerical experiment is presented to illustrate the theory.

Authors:I.Th. Famelis; Z. Jackiewicz Pages: 79 - 93 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): I.Th. Famelis, Z. Jackiewicz In this work we describe a new approach to the construction of diagonally implicit multistage integration methods (DIMSIMs) for the numerical solution of initial value problems for ordinary differential equations (ODEs). Differential Evolution, a very popular computational intelligence technique is employed to construct type 1 and type 2 methods with better or equivalent characteristics to the methods presented in the literature. The numerical results in selected problems justify this argument.

Authors:Michał Braś; Angelamaria Cardone; Zdzisław Jackiewicz; Bruno Welfert Pages: 94 - 114 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Michał Braś, Angelamaria Cardone, Zdzisław Jackiewicz, Bruno Welfert The order reduction phenomenon for general linear methods (GLMs) for stiff differential equations is investigated. It turns out that, similarly as for standard Runge–Kutta methods, the effective order of convergence for a large class of GLMs applied to stiff differential systems, is equal to the stage order of the method. In particular, it is demonstrated that the global error ‖ e [ n ] ‖ of GLMs of order p and stage order q applied to the Prothero–Robinson test problem y ′ ( t ) = λ ( y ( t ) − φ ( t ) ) + φ ′ ( t ) , t ∈ [ t 0 , T ] , y ( t 0 ) = φ ( t 0 ) , is O ( h q ) + O ( h p ) as h → 0 and h λ → − ∞ . Moreover, for GLMs with Runge–Kutta stability which are A ( 0 ) -stable and for which the stability function R ( z ) of the underlying Runge–Kutta methods, (i.e., the corresponding RK methods which have the same absolute stability properties as the GLMs), is such that R ( ∞ ) ≠ 1 , the global error satisfies ‖ e [ n ] ‖ = O ( h q + 1 ) + O ( h p ) as h → 0 and h λ → − ∞ . These results are confirmed by numerical experiments.

Authors:Zhengjie Sun; Wenwu Gao Pages: 115 - 125 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Zhengjie Sun, Wenwu Gao Based on quasi-interpolation, the paper proposes a meshless scheme for Hamiltonian PDEs with conservation properties. There are two key features of the proposed scheme. First, it is constructed from scattered sampling data. Second, it conserves energy for both linear and nonlinear Hamiltonian PDEs. Moreover, if the considered Hamiltonian PDEs additionally possess some other quadric invariants (i.e., the mass in the Schrödinger equation), then it can even preserve them. Error estimates (including the truncation error and the global error) of the scheme are also derived in the paper. To demonstrate the efficiency and superiority of the scheme, some numerical examples are provided at the end of the paper. Both theoretical and numerical results demonstrate that the scheme is simple, easy to compute, efficient and stable. More importantly, the scheme conserves the discrete energy and thus captures the long-time dynamics of Hamiltonian systems.

Authors:Zi-Cai Li; Ming-Gong Lee; Hung-Tsai Huang; John Y. Chiang Pages: 126 - 145 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Zi-Cai Li, Ming-Gong Lee, Hung-Tsai Huang, John Y. Chiang The method of fundamental solutions (MFS) was first used by Kupradze in 1963 [21]. Since then, there have appeared numerous reports of the MFS. Most of the existing analysis for the MFS are confined to Dirichlet problems on disk domains. It seems to exist no analysis for Neumann problems. This paper is devoted to Neumann problems in non-disk domains, and the new stability analysis and the error analysis are made. The bounds for both condition numbers and errors are derived in detail. The optimal convergence rates in L 2 and H 1 norms in S are achieved, and the condition number grows exponentially as the number of fundamental functions increases. To reduce the huge condition numbers, the truncated singular value decomposition (TSVD) may be solicited. Numerical experiments are provided to support the analysis made. The analysis for Neumann problems in this paper is intriguing due to its distinct features.

Authors:Zhengguang Liu; Aijie Cheng; Xiaoli Li; Hong Wang Pages: 146 - 163 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Zhengguang Liu, Aijie Cheng, Xiaoli Li, Hong Wang In this paper, we study fast Galerkin finite element methods to solve a space–time fractional diffusion equation. We develop an optimal piecewise-linear and piecewise-quadratic finite element methods for solving this problem and give optimal error estimates. Furthermore, we develop piecewise-constant discontinuous finite element method for discontinuous problem of this model. Importantly, a fast solution technique to accelerate non-square Toeplitz matrix–vector multiplications which arise from both continuous and discontinuous Galerkin finite element discretization respectively is considered. This fast solution technique is based on fast Fourier transform and depends on the special structure of coefficient matrices and it helps to reduce the computational work from O ( N 3 ) required by the traditional methods to O ( N log 2 N ) , where N is the size (number of spatial grid points) of the coefficient matrices for every time step. Moreover, the applicability and accuracy of the method are demonstrated by numerical experiments to support our theoretical analysis.

Authors:Bin Wang; Fanwei Meng; Yonglei Fang Pages: 164 - 178 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Bin Wang, Fanwei Meng, Yonglei Fang In this paper we discuss the efficient implementation of RKN-type Fourier collocation methods, which are used when solving second-order differential equations. The proposed implementation relies on an alternative formulation of the methods and their blended formulation. The features and effectiveness of the implementation are confirmed by the performance of the methods on three numerical tests.

Authors:Cheng Zhang; Hui Wang; Jingfang Huang; Cheng Wang; Xingye Yue Pages: 179 - 193 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Cheng Zhang, Hui Wang, Jingfang Huang, Cheng Wang, Xingye Yue The nonlinear stability and convergence analyses are presented for a second order operator splitting scheme applied to the “good” Boussinesq equation, coupled with the Fourier pseudo-spectral approximation in space. Due to the wave equation nature of the model, we have to rewrite it as a system of two equations, for the original variable u and v = u t , respectively. In turn, the second order operator splitting method could be efficiently designed. A careful Taylor expansion indicates the second order truncation error of such a splitting approximation, and a linearized stability analysis for the numerical error function yields the desired convergence estimate in the energy norm. In more details, the convergence in the energy norm leads to an ℓ ∞ ( 0 , T ⁎ ; H 2 ) convergence for the numerical solution u and ℓ ∞ ( 0 , T ⁎ ; ℓ 2 ) convergence for v = u t . And also, the presented convergence is unconditional for the time step in terms of the spatial grid size, in comparison with a severe time step restriction, Δ t ≤ C h 2 , required in many existing works.

Authors:Feng Liao; Luming Zhang; Shanshan Wang Pages: 194 - 212 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Feng Liao, Luming Zhang, Shanshan Wang In this article, we formulate two orthogonal spline collocation schemes, which consist of a nonlinear and a linear scheme for solving the coupled Schrödinger–Boussinesq equations numerically. Firstly, the conservation laws of our schemes are derived. Secondly, the existence solutions of our schemes are investigated. Thirdly, the convergence and stability of the nonlinear scheme are analyzed by means of discrete energy methods, while the convergence of the linear scheme is proved by cut-off function technique. Finally, numerical results are reported to verify our theoretical analysis for the numerical methods.

Authors:Zhongying Chen; Yuesheng Xu; Jiehua Zhang Pages: 213 - 224 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Zhongying Chen, Yuesheng Xu, Jiehua Zhang This paper presents a second-order hybrid finite volume method for solving the Stokes equation on a two dimensional domain. The trial function space of the method for velocity is chosen to be a quadratic conforming finite element space with a hierarchical decomposition technique on triangular meshes, and its corresponding test function space consists of piecewise constant functions and piecewise quadratic polynomial functions based on a dual partition of the domain. The trial function space and test function space of the method for pressure are chosen to be a linear finite element space. We derive the inf-sup conditions of the discrete systems of the method on triangular meshes by using a relationship between the finite volume method and the finite element method. The well-posedness of the proposed finite volume method is obtained by using the Babuska–Lax–Milgram theorem. The error estimates of the optimal order are obtained in the H 1 -norm for velocity and in the L 2 -norm for pressure. Numerical experiments are presented to illustrate the theoretical results.

Authors:Saifon Chaturantabut Pages: 225 - 238 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Saifon Chaturantabut This work presents a model reduction framework using a temporal localized basis approach to efficiently reduce the simulation time for nonlinear dynamical systems with rapid changes over a short time period, and proposes a corresponding a priori error bound. This framework employs the proper orthogonal decomposition (POD) to construct localized basis sets from different temporal subdomains, which can be used in the Galerkin projection to accurately capture the important local dynamics of the system. The discrete empirical interpolation method (DEIM) with the corresponding temporal localized basis sets is then applied to efficiently compute the projected nonlinear terms. A heuristic procedure for subdividing snapshots over the temporal domain is proposed. This procedure first partitions the set of snapshots where there are possible significant changes in system dynamics, and then uses the notion of distance between subspaces to later remove unnecessary partitioning. An a priori error bound is derived to confirm the convergence of this framework and to explain how the propagated errors from the localized reduced systems affect the overall accuracy. Numerical experiments demonstrate the accuracy improvement of the temporal localized framework through a parametrized nonlinear miscible flow simulation. The results show the applicability of the proposed approach to various parameter values that are not necessary used for generating the POD and DEIM localized basis sets.

Authors:Giovanni Capobianco; Dajana Conte; Beatrice Paternoster Pages: 239 - 247 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Giovanni Capobianco, Dajana Conte, Beatrice Paternoster It is the purpose of this paper to construct an error estimation for highly stable two-step continuous methods derived in [7], in order to use it in a variable stepsize implementation. New families of two step almost collocation methods are constructed, by using a collocation technique which permits to increase the uniform order of one step collocation methods, without increasing the computational cost and by maintaining good stability properties, thus avoiding the order reduction phenomenon. Numerical experiments confirm the effectiveness of the proposed methods.

Authors:Qingtang Jiang; Dale K. Pounds Pages: 1 - 18 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Qingtang Jiang, Dale K. Pounds Multiresolution techniques for (mesh-based) surface processing have been developed and successfully used in surface progressive transmission, compression and other applications. A triangular mesh allows 3 , dyadic and 7 refinements. The 3 -refinement is the most appealing one for multiresolution data processing since it has the slowest progression through scale and provides more resolution levels within a limited capacity. The 3 refinement has been used for surface subdivision and for discrete global grid systems. Recently lifting scheme-based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets (with either dyadic or 3 refinement) have certain smoothness, they will have big supports. In other words, the corresponding multiscale algorithms have large templates; and this is undesirable for surface processing. On the other hand, frames provide a flexibility for the construction of system generators (called framelets) with high symmetry and smaller supports. In this paper we study highly symmetric 3 -refinement wavelet bi-frames for surface processing. We design the frame algorithms based on the vanishing moments and smoothness of the framelets. The frame algorithms obtained in this paper are given by templates so that one can easily implement them. We also present interpolatory 3 subdivision-based frame algorithms. In addition, we provide frame ternary multiresolution algorithms for boundary vertices on an open surface.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Lars Grüne, Thuy T.T. Le We introduce a new formulation of the minimum time problem in which we employ the signed minimum time function positive outside of the target, negative in its interior and zero on its boundary. Under some standard assumptions, we prove the so called Bridge Dynamic Programming Principle (BDPP) which is a relation between the value functions defined on the complement of the target and in its interior. Then owing to BDPP, we obtain the error estimates of a semi-Lagrangian discretization of the resulting Hamilton–Jacobi–Bellman equation. In the end, we provide numerical tests and error comparisons which show that the new approach can lead to significantly reduced numerical errors.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Fanhai Zeng, Changpin Li In this paper, a new Crank–Nicolson finite element method for the time-fractional subdiffusion equation is developed, in which a novel time discretization called the modified L1 method is used to discretize the Riemann–Liouville fractional derivative. The present method is unconditionally stable and convergent of order O ( τ 1 + β + h r + 1 ) , where β ∈ ( 0 , 1 ) , τ and h are the step sizes in time and space, respectively, and r is the degree of the piecewise polynomial space. The derived method is reduced to the classical Crank–Nicolson method when β → 1 . The new time discretization is also used to solve the fractional cable equation. And the unconditional stability and convergence are given. Numerical examples are provided which support the theoretical analysis. The comparison with the existing methods are also given, which shows good performances of the present methods.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Armin Westerkamp, Manuel Torrilhon We present an analysis of a set of parametrized boundary conditions for a Stokes–Brinkman model in two space dimensions, discretized by finite elements. We particularly point out an instability which arises when these boundary conditions are posed on a curved line, which then leads to unphysical oscillations. In contrast to a Navier-slip condition, which is prone to Babuška's paradox, this instability can be traced back to the continuous level. We claim that the stability in these cases depend on the amount of curvature at the boundary, which is shown in a reduced setting in cylinder coordinates. The transition to a two dimensional Cartesian case is then based on numerical studies, which further substantiate the claim. Lastly, stabilization techniques are motivated that enhance the continuous FEM setting and are conveniently able to deal with arising oscillations.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Asma Toumi, Guillaume Dufour, Ronan Perrussel, Thomas Unfer We propose an asynchronous method for the explicit integration of multi-scale partial differential equations. This method is restricted by a local CFL (Courant Friedrichs Lewy) condition rather than the traditional global CFL condition. Moreover, contrary to other local time-stepping (LTS) methods, the asynchronous algorithm permits the selection of independent time steps in each mesh element. We derived an asynchronous Runge–Kutta 2 (ARK2) scheme from a standard explicit Runge–Kutta method and we proved that the ARK2 scheme is second order convergent. Comparing with the classical integration, the asynchronous scheme is effective in terms of computation time.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Ömür Kıvanç Kürkçü, Ersin Aslan, Mehmet Sezer In this study, we solve some widely-used model problems consisting of linear, nonlinear differential and integral equations, employing Dickson polynomials with the parameter-α and the collocation points for an efficient matrix method. The convergence of a Dickson polynomial solution of the model problem is investigated by means of the residual function. We encode useful computer programs for model problems, in order to obtain the precise Dickson polynomial solutions. These solutions are plotted along with the exact solutions in figures and the numerical results are compared with other well-known methods in tables.

Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): G. Califano, G. Izzo, Z. Jackiewicz We present a systematic approach to the construction of starting procedures for general linear methods (GLMs) of order p and stage order q = p . Order conditions for starting procedures based on the generalized Runge–Kutta (RK) are derived using the theory of rooted trees, elementary differentials, and elementary weights, and examples of generalized RK formulas are given up to the order p = 4 .

Authors:Mokhtary Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): P. Mokhtary The main concern of this paper is to develop and analyze an operational Tau method for obtaining the numerical solution of fractional weakly singular integro-differential equations when the Jacobi polynomials are used as natural basis functions. This strategy is an application of the matrix–vector–product approach in Tau formulation of the problem. We first study the regularity of the exact solution and show that some derivatives of the exact solution have a singularity at origin dependence on both order of fractional derivative and weakly singular kernel function which makes poor convergence results for the Tau discretization of the problem. In order to recover high-order of convergence, we propose a new variable transformation to regularize the given functions and then to approximate the solution via a satisfactory order of convergence using an operational Tau method. Convergence analysis of this novel method is presented and the expected spectral rate of convergence for the proposed method is established. Numerical results are given which confirm both the theoretical predictions obtained and efficiency of the proposed method.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): K. Šišková, M. Slodička In the present paper, we deal with an inverse source problem for a time-fractional wave equation in a bounded domain in R d . The time-dependent source is determined from an additional measurement in the form of integral over the space subdomain. The existence, uniqueness and regularity of a weak solution are obtained. A numerical algorithm based on Rothe's method is proposed, a priori estimates are proved and convergence of iterates towards the solution is established.

Authors:Laurita Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): C. Laurita In this paper we propose a new boundary integral method for the numerical solution of Neumann problems for the Laplace equation, posed in exterior planar domains with piecewise smooth boundaries. Using the single layer representation of the potential, the differential problem is reformulated as a classical boundary integral equation. The use of a smoothing transformation and the introduction of a modified Gauss–Legendre quadrature formula for the approximation of the singular integrals, which turns out to be convergent, leads us to apply a Nyström type method for the numerical solution of the integral equation. We solve some test problems and present the numerical results in order to show the efficiency of the proposed procedure.