Applied Numerical Mathematics Research Articles

An inertial three-term hybrid CG-based projection method for constrained nonlinear pseudo-monotone equations with applications

In this work, we first take some modifications to the three-term conjugate gradient (CG) search direction proposed by Liu et al. [Applied Numerical Mathematics, 2022,175: 98-107], and the new search direction satisfies the sufficient descent and trust region properties which is independent of any line search. Then, combining with the inertial extrapolation step-length and hyperplane projection technique, we propose an inertial three-term hybrid CG-based projection method to solve nonlinear pseudo-monotone equations with convex constraints. The global convergence of our proposed method is analyzed and established without the Lipschitz continuity. Some numerical experiments are executed on the standard constrained nonlinear equations and their corresponding performance profiles are illustrated, which show that our proposed method is feasible and effective. Finally, our proposed method is extended to restore the blurred images contaminated by the Gaussian noise.

How Does Governance Modify the Relationship between Public Finance and Economic Growth: A Global Analysis

How Does Governance Modify the Relationship between Public Finance and Economic Growth: A Global Analysis

Algebraic construction of a third order difference approximations for fractional derivatives and applications

Finite difference approximations for fractional derivatives based on Grunwald formula are well known to be of first order accuracy, but display unstable solutions with known numerical methods. The shifted form of the Grunwald approximation removes this instability and keeps the same first order accuracy. Higher order approximations have been obtained by convex combinations of various shifted Grunwald approximations. Recently, a second order shifted Grunwald type approximation was constructed algebraically through a generating function. In this paper, we derive a new third order approximation from this second order approximation by preconditioning the fractional differential operator. This approximation is used with Crank-Nicolson numerical scheme to approximate the solutions of space-fractional diffusion equations by the same preconditioning. Stability and convergence of the numerical scheme are analysed, supported by numerical results showing third order convergence. References Boris Baeumer, Mihaly Kovacs, and Harish Sankaranarayanan. Higher order grunwald approximations of fractional derivatives and fractional powers of operators. Transactions of the American Mathematical Society, 367(2):813–834, 2015. doi:10.1090/S0002-9947-2014-05887-X E. Barkai, R. Metzler, and J. Klafter. From continuous time random walks to the fractional fokker-planck equation. Physical Review E, 61(1):132, 2000. doi:10.1103/PhysRevE.61.132 Z. Hao, Z. Sun, and W. Cao. A fourth-order approximation of fractional derivatives with its applications. Journal of Computational Physics, 281:787–805, 2015. doi:10.1016/j.jcp.2014.10.053 Ch Lubich. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3):704–719, 1986. doi:10.1137/0517050 M. M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65–77, 2004. doi:10.1016/j.cam.2004.01.033 H. M. Nasir, B. L. K. Gunawardana, and H. M. N. P. Aberathna. A second order finite difference approximation for the fractional diffusion equation. International Journal of Applied Physics and Mathematics, 3(4):237, 2013. doi:10.7763/IJAPM.2013.V3.212 H. M. Nasir and K. Nafa. A new second order approximation for fractional derivatives with applications. SQU Journal of Science, 23(1):43–55, 2018. doi:10.24200/squjs.vol23iss1pp43-55 W. Tian, H. Zhou, and W. Deng. A class of second order difference approximations for solving space fractional diffusion equations. Mathematics of Computation, 84(294):1703–1727, 2015. doi:10.1090/S0025-5718-2015-02917-2 Y. Yu, W. Deng, and Y. Wu. Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations. arXiv preprint arXiv:1408.6364, 2014. doi:10.4310/CMS.2017.v15.n5.a1 Y. Yu, W. Deng, Y. Wu, and J. Wu. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Applied Numerical Mathematics, 112:126–145, 2017. doi:10.1016/j.apnum.2016.10.01 L. Zhao and W. Deng. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numerical Methods for Partial Differential Equations, 31(5):1345–1381, 2015. doi:10.1002/num.21947

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries

In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular flow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu (2006) for the fractional diffusion-wave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics 56 (2006) 193–209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional step-strain, we observe the different rates of stress relaxation obtained with different values of α and β (the fractional order exponents that regulate the viscoelastic response of the complex fluids).

Preconditioning of a Coupled Cahn-Hilliard Navier-Stokes System

Recently, Garcke et al.[Garcke, Hinze, Kahle, A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Applied Numerical Mathematics 99, pp. 151-171, 2016] developed a consistent discretization scheme for a thermodynamically consistent diffuse interface model for incompressible two-phase flows with different densities. At the heart of this method lies the solution of large and sparse linear systems that arise in a semismooth Newton method. We propose the use of preconditioned Krylov subspace solvers using effective Schur complement approximations. Numerical results illustrate the efficiency of our approach. In particular, our preconditioner is shown to be robust with respect to parameter changes.

Adjoint-based hp -adaptivity on anisotropic meshes for high-order compressible flow simulations

Abstract We present an efficient adjoint-based hp -adaptation methodology on anisotropic meshes for high order Discontinuous Galerkin schemes applied to (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. The refinement strategy is based on a recently proposed interpolation error estimate [Dolejsi V. Anisotropic hp-adaptive method based on interpolation error estimates in the lq-norm. Applied Numerical Mathematics 2014;82:80-114.], as well as an adjoint-based error estimate. Using the two error estimates, we determine the size and the shape of the triangular mesh elements on the desired mesh to be used for the subsequent adaptation steps. This is done using the concept of mesh-metric duality, where metric tensors encode information about mesh elements. The metric tensors that correspond to the desired mesh elements can be passed to a metric-conforming mesh generator to create the required anisotropic mesh. The effectiveness of the adaptation methodology is demonstrated using numerical experiments, involving a scalar model problem with a strong boundary layer, as well as various compressible flow test cases.

A Moving-Mesh Finite Element Method and its Application to the Numerical Solution of Phase-Change Problems

A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase. The algorithm uses a distributed conservation principle to de- termine nodal mesh velocities, which are then used to move the nodes. The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation, which represents a gener- alisation of the original algorithm presented in Applied Numerical Mathematics, 54:450–469 (2005). Having described the details of the generalised algorithm it is validated on two test cases from the original paper and is then applied to one-phase and, for the first time, two- phase Stefan problems in one and two space dimensions, paying particular attention to the implementation of the interface boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness of the method, including comparisons against analytical solutions where available.

Kronecker product approximations for image restoration with new mean boundary conditions

As an alternative to classic boundary conditions, new mean boundary conditions for image restoration problem have been recently introduced by Shi and Chang [Y. Shi, Q. Chang, Acceleration methods for image restoration problem with different boundary conditions, Applied Numerical Mathematics 58 (2008) 602–614]. In this paper, we propose an efficient scheme for computing the Kronecker product approximations of blurring matrices with the new mean boundary conditions. Our scheme does not require the symmetry condition of point spread functions. Detailed experiments in image restoration are given to demonstrate the efficiency of our scheme.

Rigorous model-based safety analysis for nonlinear continuous-time systems

A method is presented for the quantitative, model-based safety analysis of nonlinear continuous-time hybrid systems. This method uses the region-transition-model (RTM) framework of [Huang, H., Adjiman, C. S., & Shah, N. (2002). Quantitative framework for reliable safety analysis. AIChE Journal, 48, 78–96], together with a recently developed technique [Lin, Y., & Stadtherr, M. A. (2007). Validated solutions of initial value problems for parametric ODEs. Applied Numerical Mathematics, 57, 1145–1162] for the rigorous global analysis of nonlinear, continuous-time systems with uncertain initial conditions and/or parameters. Given an operating region described by bounds on possible initial conditions, inputs and model parameters, and a finite time horizon, the method can determine which operating subregions lead to safe operation. Numerical examples are presented that demonstrate the effectiveness of the method. This approach can supplement and complement the more qualitative techniques that are widely used for hazard identification and safety analysis.

A posteriori parameter choice with an efficient discretization scheme for solving ill-posed problems

Recently, Rajan [M.P. Rajan, An efficient discretization scheme for solving the ill-posed problems, Journal of Mathematical Analysis and Applications 313 (2) (2006) 654–677] considered a discretization scheme for solving ill-posed problems and obtained optimal error estimates under an a priori parameter choice strategy with a particular smoothness assumption on the solution. It is shown that the computational information for solving the system is far less than the traditional projection schemes. Neubauer [A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error, Applied Numerical Mathematics 4 (1988) 507–519] and Engl and Gfrerer [H.W. Engl, H. Gfrerer, A posteriori parameter choice for general regularization methods for solving ill-posed problems, Applied Numerical Mathematics 4 (1988) 395–417] suggested an a posteriori method for choosing the regularization parameter which does not require any knowledge about the exact solution, respectively under finite dimensional as well as general setting. In this paper, we consider an a posteriori parameter choice for finite dimensional approximation which does not require any information about the smoothness of the exact solution and apply to the discretization scheme considered by Rajan. The computational efficiency of the scheme is illustrated through numerical examples.

A partitioning algorithm for block-diagonal matrices with overlap

We present an algorithm for partitioning a sparse matrix into a matrix that has blocks on the diagonal such that two consecutive blocks can overlap. We refer to this form of the matrix as block diagonal matrix with overlap. The partitioned matrix is suitable for applying the explicit formulation of multiplicative Schwarz (EFMS) [G.A. Atenekeng Kahou, E. Kamgnia, B. Philippe, An explicit formulation of the multiplicative Schwarz preconditioner, Journal of Applied Numerical Mathematics 57 (2007) 1197–1213] used as a preconditioner for solving a sparse unsymmetric system of linear equations Ax = b . The proposed algorithm partitions the graph of the matrix A into k parts such that every part V i has connecting edges with at most two neighbors V i - 1 and V i + 1 . First, an ordering algorithm that reduces the profile of the matrix, and an initial block-diagonal partition with overlap is obtained. Second, an iterative strategy is used to further refine the partitioning by allowing vertices to be moved between partitions. Experiments performed on real-world matrices show the usefulness of this approach.

Limiters for high-order discontinuous Galerkin methods

We describe a limiter for the discontinuous Galerkin method that retains as high an order as possible, and does not automatically reduce to first order. The limiter is a generalization of the limiter introduced in [R. Biswas, K. Devine, J.E. Flaherty, Parallel adaptive finite element methods for conservation laws, Applied Numerical Mathematics 14 (1994) 255–284]. We present the one-dimensional case and extend it to two-dimensional problems on tensor-product meshes. Computational results for examples with both smooth and discontinuous solutions are shown.

An improved and generalized second order, unconditionally positive, mass conserving integration scheme for biochemical systems

Bruggeman et al. [J. Bruggeman, H. Burchard, B. Kooi, B. Sommeijer, A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems, Applied Numerical Mathematics 57 (1) (2007) 36–58] presented novel first and second-order implicit integration schemes which guarantee both conservation (in a strict biochemical sense) and positive-definite results (hereafter, BBKS1 and BBKS2 respectively). In this paper we show that it is possible to achieve substantially more accurate results by making a minor modification to the BBKS-schemes (hereafter, we refer to the revised first- and second-order schemes as mBBKS1 and mBBKS2). The BBKS and the mBBKS schemes are shown to be special cases of a more generalized scheme (dubbed gBBKS). All operate by automatically slowing the forecaster time-step-averaged reaction rates in order to maintain positivity. The mBBKS scheme induces less slowing than the BBKS one. With a second modification, gBBKS-type schemes can become unusual adaptive-time-step schemes. Unfortunately, for the ODE-systems that we examined, the adaptive-mBBKS variant proves to be substantially less efficient than standard adaptive schemes (in this instance, adaptive time-step second-order explicit Runge–Kutta). Nonetheless, it is possible that the adaptive-mBBKS-scheme would become more competitive when the right-hand sides of a system of ODEs are more expensive to evaluate.

Nonstandard finite-difference schemes for the Lotka–Volterra systems: generalization of Mickens's method

Mickens [Applied Numerical Mathematics, 45, pp. 309–314, 2003] constructed a nonstandard finite difference scheme for the Lotka–Volterra predator–prey system which, for positive initial conditions, gives rise to periodic solutions. His method amazingly preserves the oscillation feature of the Lotka–Volterra system while many numerical methods give solutions that spiral into or out of the positive-valued fixed-point. The reason is that Mickens's scheme is noncanonically symplectic. Applying the same idea, we generalize Mickens's method and produce a class of nonstandard symplectic numerical methods for the above Lotka–Volterra system. These methods are all symplectic with respect to a noncanonical symplectic structure. They all have the property that the computed points do not spiral. Hence, these methods also preserve the oscillation feature of the predator–prey system.

Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton–Jacobi equations

In this note, we reinterpret a discontinuous Galerkin method originally developed by Hu and Shu [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690] (see also [O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434]) for solving Hamilton–Jacobi equations. With this reinterpretation, numerical solutions will automatically satisfy the curl-free property of the exact solutions inside each element. This new reinterpretation allows a method of lines formulation, which renders a more natural framework for stability analysis. Moreover, this reinterpretation renders a significantly simplified implementation with reduced cost, as only a smaller subspace of the original solution space in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is used and the least squares procedure used in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is completely avoided.

MULTI-DIMENSIONAL SEMI-LAGRANGIAN CHARACTERISTIC APPROACH TO THE SHALLOW WATER EQUATIONS BY THE CIP METHOD

International Journal of Computational Engineering ScienceVol. 05, No. 03, pp. 699-730 (2004) No AccessMULTI-DIMENSIONAL SEMI-LAGRANGIAN CHARACTERISTIC APPROACH TO THE SHALLOW WATER EQUATIONS BY THE CIP METHODYOICHI OGATA and TAKASHI YABEYOICHI OGATADepartment of Mechanical Engineering and Science, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo, 152-8552, Japan Search for more papers by this author and TAKASHI YABEDepartment of Mechanical Engineering and Science, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo, 152-8552, Japan Search for more papers by this author https://doi.org/10.1142/S1465876304002642Cited by:10 Previous AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractWe propose a multi-dimensional scheme to solve the shallow water equations by using the CIP(Constrained Interpolation Profile/Cubic Interpolated Pseudoparticle) method which has the accuracy by one-order of magnitude better than the cubic Lagrange. This third-order accuracy is kept even in non-uniform grid, while the cubic Lagrange becomes the first order. The semi-Lagrangian approach to the characteristic equations has been used for solving gravity waves because it can make time step be much longer than the Courant-Friedrichs-Lewy (CFL) condition. Even in two dimensions, the method can give a symmetrical wave propagation under the large CFL condition. References T. Yabe, F. Xiao and T. Utsumi, J. Comput. Phys. 169, 556 (2001). Crossref, Google ScholarF. H. Harlow and J. E. Welch, Phys. Fluids 8, 2182 (1965). Crossref, Google Scholar I. Kinnmark , The Shallow Water Wave Equations; Formulation, Analysis and Application ( Springer-Verlag , Berlin Heidelberg New York Tokyo , 1986 ) . Crossref, Google ScholarG. Erbes, Mon. Wea. Rev. 121, 3443 (1993). Crossref, Google ScholarH. Takewaki, A. Nishiguchi and T. Yabe, J. Comput. Phys. 61, 261 (1985). Crossref, Google ScholarT. Yabe and E. Takei, J. Phys. Soc. Japan 57, 2598 (1988). 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Crossref, Google Scholar FiguresReferencesRelatedDetailsCited By 10Bicompact interpolation-characteristic scheme of the third order of approximation for the linear transport equationBoris Vadimovich Rogov1 January 2020 | Keldysh Institute Preprints, No. 106CIP Method of Characteristics for the Solution of Tide Wave EquationsYafei Nie, Kai Fu and Xianqing Lv2 Jul 2018 | Advances in Mathematical Physics, Vol. 2018Tree-based mesh-refinement GPU-accelerated tsunami simulator for real-time operationMarlon Arce Acuña and Takayuki Aoki21 September 2018 | Natural Hazards and Earth System Sciences, Vol. 18, No. 9New optimized fourth-order compact finite difference schemes for wave propagation phenomenaMaurizio Venutelli1 Jan 2015 | Applied Numerical Mathematics, Vol. 87Numerical Study of Violent Impact Flow Using a CIP-Based ModelQiao-ling Ji, Xi-zeng Zhao and Sheng Dong1 Jan 2013 | Journal of Applied Mathematics, Vol. 2013Application of the characteristic CIP method to a shallow water model on the sphereXindong Peng, Yan Chang, Xingliang Li and Feng Xiao19 June 2010 | Advances in Atmospheric Sciences, Vol. 27, No. 4Conservative semi-Lagrangian CIP technique for the shallow water equationsTakashi Yabe and Youichi Ogata7 November 2009 | Computational Mechanics, Vol. 46, No. 1Multi-dimensional conservative semi-Lagrangian method of characteristics CIP for the shallow water equationsKunihiko Toda, Youichi Ogata and Takashi Yabe1 Jul 2009 | Journal of Computational Physics, Vol. 228, No. 13Implementation of a non-oscillatory and conservative scheme into magnetohydrodynamic equationsShin Tanaka, Takayuki Umeda, Yosuke Matsumoto, Takahiro Miyoshi and Tatsuki Ogino31 August 2009 | Earth, Planets and Space, Vol. 61, No. 7Accuracy study of the IDO scheme by Fourier analysisYohsuke Imai and Takayuki Aoki1 Sep 2006 | Journal of Computational Physics, Vol. 217, No. 2 Recommended Vol. 05, No. 03 Metrics History PDF download

Skew-Hermitian Triangular Splitting Iteration Methods for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts

By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.

Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case.

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.

New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows.

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high-speed high-altitude aerodynamics and the development of micrometre-sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non-equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non-equilibrium flows.

Time discretization of vibro‐impact

We consider vibro–impact problems, i.e. mechanical systems with a finite number of degrees of freedom submitted to perfect unilateral constraints. The dynamics is basically described by a second–order measure differential inclusion for the unknown position completed with a constitutive impact law. Another formulation of the problem as a frictionless sweeping process is possible: the unknown velocity belongs to an appropriate functional space and satisfies a first order measure differential inclusion. The equivalence of these two formulations is studied. They lead to time–discretizations written in terms of positions or in terms of velocities, respectively. We present these different schemes and we compare them on the simple test–problem of a bouncing ball. We recall the convergence results in the single constraint case. Moreover, an example of implementation of the scheme derived from the basic description of the dynamics is presented. Finally, in the multi–constraint case, we point out some theoretical and computational difficulties.