994 results
Search Results
2. A survey on the high convergence orders and computational convergence orders of sequences.
- Author
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Cătinaş, Emil
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STOCHASTIC convergence , *COMPUTATIONAL mathematics , *ASYMPTOTIC distribution , *NONLINEAR equations , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis - Abstract
Abstract Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the Q -convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention. The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago. Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations. Tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
3. Robust finite-time guidance against maneuverable targets with unpredictable evasive strategies.
- Author
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Zhang, Ran, Wang, Jiawei, Li, Huifeng, Li, Zhenhong, and Ding, Zhengtao
- Subjects
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ROBUST control , *PROBLEM solving , *STABILITY theory , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the interceptor and the target, but also by external un-modeled factors. By robustly stabilizing a line-of-sight rate, this paper proposes an integrated continuous finite-time disturbance observer/bounded continuous finite-time stabilizer strategy. The design of this integrated strategy has two points: 1) effect of a target maneuver is modeled as disturbance and then is estimated by the second-order homogeneous observer; 2) the finite-time stabilizer is actively coupled with the observer. Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are discussed. Numerical comparison studies demonstrate the guidance performance. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
4. A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations.
- Author
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Tian, Tian, Zhai, Qilong, and Zhang, Ran
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DEGREES of freedom , *GALERKIN methods , *STOKES equations , *NUMERICAL analysis , *STOCHASTIC convergence - Abstract
In this paper, a modified weak Galerkin method is proposed for the Stokes problem. The numerical scheme is based on a novel variational form of the Stokes problem. The degree of freedoms in the modified weak Galerkin method is less than that in the original weak Galerkin method, while the accuracy stays the same. In this paper, the optimal convergence orders are given and some numerical experiments are presented to verify the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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5. Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes.
- Author
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Grishagin, Vladimir, Israfilov, Ruslan, and Sergeyev, Yaroslav
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STOCHASTIC convergence , *NUMERICAL analysis , *GLOBAL optimization , *COMPUTER algorithms , *MATHEMATICAL domains - Abstract
This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursive way. The second approach as a reduction scheme uses Peano-type space-filling curves mapping multidimensional domains onto one-dimensional intervals. In the frameworks of both the approaches, several univariate algorithms belonging to the characteristical class of optimization techniques are used for carrying out the one-dimensional optimization. Theoretical part of the paper contains a substantiation of global convergence for the considered methods. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator used broadly for testing global optimization algorithms. Results for representative problem sets of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme with respect to other tested methods. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. Convergence and error estimates of viscosity-splitting finite-element schemes for the primitive equations.
- Author
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Guillén-González, F. and Redondo-Neble, M.V.
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STOCHASTIC convergence , *ERROR analysis in mathematics , *FINITE element method , *NUMERICAL analysis , *NONLINEAR differential equations - Abstract
This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O ( k + h l ) for l = 1 or l = 2 considering either first or second order finite-element approximations ( k and h being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint k ≤ C h 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2017
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7. Consensus analysis of directed multi-agent networks with singular configurations.
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Wu, Yonghong and Guan, Zhi-Hong
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ARTIFICIAL neural networks , *CONTINUOUS time systems , *MULTIAGENT systems , *NUMERICAL analysis , *DERIVATIVES (Mathematics) , *STOCHASTIC convergence - Abstract
This paper discusses the continuous-time consensus problems for directed multi-agent networks under certain coupling or control. Since agents are driven not only by their neighbors' states but also by their derivatives in many realistic situations, dynamical networks described by singular systems are appropriate to study. Consensus problems for such multi-agent networks are considered when the agents communicate in the presence or absence of time delays. The maximum tolerated time-delay is obtained when the multi-agent network asymptotically reaches consensus. The results of this paper indicate that such multi-agent networks can achieve consensus with a demanding convergent speed through agents' interactions. Numerical examples are given to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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8. Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures.
- Author
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Huang, Yunqing, Li, Jichun, and Yang, Wei
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NUMERICAL analysis , *NANOSTRUCTURES , *TIME-domain analysis , *MAXWELL equations , *PARTIAL differential equations , *FINITE element method , *STOCHASTIC convergence - Abstract
In this paper, we discuss the time-domain Maxwell’s equations coupled to another partial differential equation, which arises from modeling of light and structure interaction at the nanoscale. One major contribution of this paper is that the well-posedness is rigorously justified for the first time. Then we propose a fully-discrete finite element method to solve this model. It is interesting to note that we need use curl conforming, divergence conforming, and L 2 finite elements for this model. Numerical stability and optimal error estimate of the scheme are proved. Numerical results justifying our theoretical convergence rate are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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9. A general implicit direct forcing immersed boundary method for rigid particles.
- Author
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Tschisgale, Silvio, Kempe, Tobias, and Fröhlich, Jochen
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PARTICULATE matter , *MULTIPHASE flow , *NUMERICAL analysis , *STOCHASTIC convergence , *FLUID dynamics - Abstract
This paper presents a new immersed boundary method for rigid particles of arbitrary shape and arbitrary density, which can be exactly zero. Especially in the latter case, the coupling of the fluid and the solid part requires special numerical techniques to obtain stability. Exploiting the direct forcing approach an algorithm with strong coupling between fluid and particles is developed, which is exempt from any global iteration between the fluid part and the solid part within a single time step. Starting point is a previously proposed method restricted to spherical particles [37]. It is proved that the coupling concept can be generalized to rigid particles with complex shapes, which is not obvious a priori . The extension requires various non-trivial methodological extensions, especially with respect to the angular motion of the particle. Using analytical techniques it is demonstrated that the implicit treatment of the coupling force in the equations of motion results in additional terms related to a surrounding numerical fluid layer. As a main improvement over other non-iterative methods the proposed scheme is unconditionally stable for the entire range of density ratios and particle shapes allowing large time steps, with Courant numbers around unity. To date, no other non-iterative coupling approach offers such a generality regarding fluid-particle interactions. In addition to the detailed description of the underlying methodology and its differences from other methods, the paper provides all modifications required to improve immersed boundary methods for particulate flows from weak to strong coupling. Furthermore, an extensive validation of the scheme for particles of different density ratios and shapes is presented. The accuracy of the method as well as the convergence behavior are assessed by systematic studies. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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- View/download PDF
10. An incremental pressure correction finite element method for the time-dependent Oldroyd flows.
- Author
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Liu, Cui and Si, Zhiyong
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FINITE element method , *DISCRETE systems , *ERROR analysis in mathematics , *STOCHASTIC convergence , *STABILITY theory - Abstract
Abstract In this paper, we present an incremental pressure correction finite element method for the time-dependent Oldroyd flows. This method is a fully discrete projection method. As we all know, most projection methods have been studied without space discretization. Then the ensuing analysis may not extend to this case. We also give the stability analysis and the optimal error analysis. The analysis is based on a time discrete error and a spatial discrete error. In order to show the effectiveness of the method, we also present some numerical results. The numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time-dependent Oldroyd flows. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
11. Parameter-uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction–diffusion equations with boundary and interior layers.
- Author
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Rao, S. Chandra Sekhara and Chawla, Sheetal
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REACTION-diffusion equations , *BOUNDARY layer equations , *COUPLED mode theory (Wave-motion) , *NUMERICAL analysis , *STOCHASTIC convergence , *PERTURBATION theory - Abstract
Abstract In this paper, we consider a coupled system of m (≥ 2) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. Highlights • A coupled system of m (≥ 2) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
12. One-leg methods for nonlinear stiff fractional differential equations with Caputo derivatives.
- Author
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Zhou, Yongtao and Zhang, Chengjian
- Subjects
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NONLINEAR differential equations , *FRACTIONAL differential equations , *STOCHASTIC convergence , *DERIVATIVES (Mathematics) , *NUMERICAL analysis - Abstract
Highlights • A type of extended one-leg methods are constructed for a class of nonlinear stiff fractional differential equations. • Under some suitable conditions, the extended one-leg methods are proved to be stable and convergent of order min { p , 2 − γ }. • Several interesting numerical examples are presented to illustrate the computational efficiency and accuracy of the extended one-leg methods. Abstract This paper is concerned with numerical solutions for a class of nonlinear stiff fractional differential equations (SFDEs). By combining the underlying one-leg methods with piecewise linear interpolation, a type of extended one-leg methods for nonlinear SFDEs with γ -order (0 < γ < 1) Caputo derivatives are constructed. It is proved under some suitable conditions that the extended one-leg methods are stable and convergent of order min { p , 2 − γ } , where p is the consistency order of the underlying one-leg methods. Several numerical examples are given to illustrate the computational efficiency and accuracy of the methods. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
13. A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem.
- Author
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Dai, Ping-Fan, Li, Jicheng, Bai, Jianchao, and Qiu, Jinming
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ITERATIVE methods (Mathematics) , *LINEAR complementarity problem , *STOCHASTIC convergence , *NUMERICAL analysis , *PROBLEM solving - Abstract
Abstract In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M -matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
14. Modified alternately linearized implicit iteration method for M-matrix algebraic Riccati equations.
- Author
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Guan, Jinrui
- Subjects
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LINEAR systems , *ITERATIVE methods (Mathematics) , *RICCATI equation , *MATRICES (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
Abstract Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
15. A new conjugate gradient method based on Quasi-Newton equation for unconstrained optimization.
- Author
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Li, Xiangli, Shi, Juanjuan, Dong, Xiaoliang, and Yu, Jianglan
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CONJUGATE gradient methods , *MATHEMATICAL optimization , *QUASI-Newton methods , *STOCHASTIC convergence , *NUMERICAL analysis , *SPECTRAL geometry - Abstract
Abstract The spectral conjugate gradient method is an effective method for large-scale unconstrained optimization problems. In this paper, based on Quasi-Newton direction and Quasi-Newton equation, a new spectral conjugate gradient method is proposed. This method is motivated by the three-term modified Polak-Ribiyre-Polyak (PRP) method and spectral parameters. The global convergence of algorithm is proved for general functions under a strong Wolfe line search. Numerical results show that the new algorithm is superior to the three-term modified PRP method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
16. Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients.
- Author
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Zhou, Shaobo and Jin, Hai
- Subjects
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STOCHASTIC convergence , *APPROXIMATION theory , *NUMERICAL analysis , *STOCHASTIC differential equations , *EULER characteristic - Abstract
Abstract In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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17. A detail preserving variational model for image Retinex.
- Author
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Gu, Zhihao, Li, Fang, and Lv, Xiao-Guang
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PROBLEM solving , *MATHEMATICAL variables , *STOCHASTIC convergence , *MATHEMATICAL regularization , *DECOMPOSITION method , *NUMERICAL analysis - Abstract
Highlights • A new variational model for image retinex is proposed. • The existence of minimizer for the variational model is proved. • An efficient algorithm is derived to solve the model by adding auxiliary variables. • The convergence of the numerical algorithm is proved under some assumptions. • The proposed method can be extended to other regularization and fidelity terms. Abstract In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
18. Unconditional L∞ convergence of a conservative compact finite difference scheme for the N-coupled Schrödinger–Boussinesq equations.
- Author
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Liao, Feng, Zhang, Luming, and Wang, Tingchun
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STOCHASTIC convergence , *FINITE differences , *SCHRODINGER equation , *BOUSSINESQ equations , *NUMERICAL analysis - Abstract
Abstract In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O (τ 2 + h 4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
19. Stability and superconvergence of efficient MAC schemes for fractional Stokes equation on non-uniform grids.
- Author
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Li, Xiaoli, Rui, Hongxing, and Chen, Shuangshuang
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STOCHASTIC convergence , *STOKES equations , *GRIDS (Cartography) , *CAPUTO fractional derivatives , *NUMERICAL analysis - Abstract
Abstract In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L 2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H 1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
20. On a second order scheme for space fractional diffusion equations with variable coefficients.
- Author
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Vong, Seakweng and Lyu, Pin
- Subjects
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FRACTIONAL differential equations , *COEFFICIENTS (Statistics) , *MATHEMATICAL physics , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
Abstract We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by further study on the generating function of the discretization matrix, second order convergence of the scheme is proved for diffusion coefficients satisfying a certain condition but are not necessary to be proportional. The theoretical results are justified by numerical tests. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. The time-dependent generalized membrane shell model and its numerical computation.
- Author
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Shen, Xiaoqin, Jia, Junjun, Zhu, Shengfeng, Li, Haoming, Bai, Lin, Wang, Tiantian, and Cao, Xiaoshan
- Subjects
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NUCLEAR shell theory , *DISCRETIZATION methods , *NUMERICAL analysis , *STOCHASTIC convergence , *ELASTODYNAMICS - Abstract
Abstract In this paper, we discuss the time-dependent generalized membrane shell model, which has not been addressed numerically in literature. We show that the solution of this model exists and is unique. We first provide a numerical method for the time-dependent generalized membrane shell. Concretely, we semi-discretize the space variable and fully discretize the problem using time discretization by the Newmark scheme. The corresponding numerical analyses of existence, uniqueness, stability and convergence with a priori error estimates are given. Finally, we present numerical experiments with a portion of the conical shell and a portion of the hyperbolic shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme. Highlights • Existence and uniqueness of the time-dependent generalized membrane shell. • Newmark scheme for the time-dependent generalized membrane shell. • Analyses of existence, uniqueness, stability, convergence and a priori error estimate for elastodynamic problem. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
22. A novel fast overrelaxation updating method for continuous-discontinuous cellular automaton.
- Author
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Yan, Fei, Pan, Peng-Zhi, Feng, Xia-Ting, Li, Shao-Jun, and Jiang, Quan
- Subjects
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ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis , *CELLULAR automata , *OPTIMAL control theory - Abstract
Highlights • A fast successive relaxation updating method for continuous-discontinuous cellular automaton(CDCA) is proposed. • A fast CDCA is developed, and increments of displacement and nodal force are enlarged by the accelerating factor. • A new discontinuity tracking method which combines cell space cutting and cell neighbor searching is proposed. • The optimal value of the accelerating factor is studied, and an adaptive iteration scheme is proposed. Abstract Because of its local property, cellular automaton method has been widely applied in different subjects, but the main problem is that the cellular updating is time-consuming. In order to improve its calculation efficiency, a fast successive relaxation updating method is proposed in this paper. Firstly, an accelerating factor ω is defined, and a fast successive relaxation updating theory and its corresponding convergence conditions are developed. In each updating step, the displacement increment is enlarged ω times as a new increment to replace the old one, similarly, the nodal forces for its neighbors caused by this displacement increment are also enlarged by the same accelerating factor, and do those updating operations until the convergence is achieved. By this method, the convergence rate is greatly improved, by a suitable accelerating factor, 90 to 98% of iteration steps are decreased compared to that of the traditional method. Besides, the influence factors for the accelerating factor are studied, and numerical studies show that the suitable accelerating factor is 1.85 < ω < 1.99, which is greatly influenced by cell stiffness, and the optimal accelerating factor is 1.96 < ω < 1.99. Finally, numerical examples are given to illustrate that the present method is effective and high convergence rate compared to the traditional method. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
23. A decoupling penalty finite element method for the stationary incompressible MagnetoHydroDynamics equation.
- Author
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Deng, Jien and Si, Zhiyong
- Subjects
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FINITE element method , *NUMERICAL analysis , *MAGNETOHYDRODYNAMICS , *STOCHASTIC convergence , *FLUID flow - Abstract
Highlights • A penalty finite element method for the steady MHD equations was given. • The solution of the penalty method convergence the solution of the steady MHD equations. • The stability analysis shows our method is stable. • The error estimate shows our method has an optimal convergence order. • The numerical results of the Hartmann flow was shown. Abstract In this paper, we give a penalty finite element method for the steady MHD equations. In this method, we decouple the MHD into two equations, one for the velocity and magnetic (u , B) , the other for the pressure p. We prove the existence of the penalty method and the optimal error estimate. Then, we give the penalty finite element method for the MHD equations. The stability analysis shows our method is stable, and the error estimate shows our method has an optimal convergence order. Finally, we give some numerical results of exact solution equation and Hartmann flow. The numerical results show that our penalty finite element method is effect. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
24. Strong convergence of a tamed theta scheme for NSDDEs with one-sided Lipschitz drift.
- Author
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Tan, Li and Yuan, Chenggui
- Subjects
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STOCHASTIC convergence , *LIPSCHITZ spaces , *NUMERICAL analysis , *PROBLEM solving , *MATHEMATICAL complexes - Abstract
This paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one-sided Lipschitz drift. Strong convergence rate is revealed under a global one-sided Lipschitz condition, while for a local one-sided Lipschitz condition, the tamed theta scheme is modified to ensure the well-posedness of implicit numerical schemes, then we show the convergence of the numerical solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
25. Implicit-explicit one-leg methods for nonlinear stiff neutral equations.
- Author
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Tan, Zengqiang and Zhang, Chengjian
- Subjects
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NONLINEAR equations , *STOCHASTIC convergence , *COMPUTATIONAL complexity , *NUMERICAL analysis , *DIFFERENTIAL equations - Abstract
In this paper, by adapting the underlying implicit-explicit (IMEX) one-leg methods (cf. [1, 2]), a class of extended IMEX one-leg (EIEOL) methods are suggested for solving nonlinear stiff neutral equations (SNEs). It is proven under some suitable conditions that EIEOL methods are D-convergent of order 2 and stable for nonlinear SNEs. Several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of EIEOL methods. Moreover, a comparison with the fully implicit one-leg methods is presented, which shows that EIEOL methods have the higher computational efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
26. Model parametrization strategies for Newton-based acoustic full waveform inversion.
- Author
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Anagaw, Amsalu Y. and Sacchi, Mauricio D.
- Subjects
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WAVE analysis , *PARAMETER estimation , *SIGNAL-to-noise ratio , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
Full waveform inversion (FWI) is an efficient engine for estimating subsurface model parameters. FWI is often implemented via local optimization methods such as conjugate gradients or steepest descent. In this paper, we examine the effect of model parameterization on the estimation of velocity models obtained via Newton-based optimization FWI methods. We consider three model parameterizations for acoustic FWI. These include velocity, slowness and slowness squared parameterizations. We numerically demonstrate that model parametrization significantly influences the convergence rate of the Newton method. A high convergence rate for Newton-based FWI algorithms can be attained by using a slowness squared model parameterization. The latter is true for data with relatively high signal-to-noise ratio. However, in the case of data contaminated by high levels of noise, the slowness parameterization provides a good trade-off between the resolution of the reconstructed model parameters and convergence rate in comparison to velocity or slowness squared parameterization. Numerical results were conducted with the Marmousi velocity model to highlight our premises. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
27. Shape optimization of conductive-media interfaces using an IGA-BEM solver.
- Author
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Kostas, K.V., Fyrillas, M.M., Politis, C.G., Ginnis, A.I., and Kaklis, P.D.
- Subjects
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BOUNDARY element methods , *ISOGEOMETRIC analysis , *NUMERICAL analysis , *HEAT conduction , *STEADY state conduction , *STOCHASTIC convergence - Abstract
In this paper, we present a method that combines the Boundary Element Method (BEM) with IsoGeometric Analysis (IGA) for numerically solving the system of Boundary Integral Equations (BIE) arising in the context of a 2-D steady-state heat conduction problem across a periodic interface separating two conducting and conforming media. Our approach leads to a fast solver with high convergence rate when compared with low-order BEM. Additionally, an optimization framework comprising a parametric model for the interface’s shape, our IGA-BEM solver, and evolutionary and gradient-based optimization algorithms is developed and tested. The optimization examples demonstrate the efficiency of the framework in generating optimum interfaces for maximizing heat transfer under various geometric constraints. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
28. Effects of helix angle and multi-mode on the milling stability prediction using full-discretization method.
- Author
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Zhou, Kai, Zhang, Jianfu, Xu, Chao, Feng, Pingfa, and Wu, Zhijun
- Subjects
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DISCRETIZATION methods , *MILLS & mill-work , *NUMERICAL analysis , *STOCHASTIC convergence , *MACHINING - Abstract
Abstract Regenerative chatter has a negative effect on the quality of machined surfaces in milling and it is thus vital to predict this accurately. The full-discretization method (FDM) is extensively utilized to predict the chatter stability. However, the effects of helix angle and multi-mode usually cause poor prediction by FDM and are often ignored. The existing fourth-order FDM has better computational efficiency and convergence rate than other existing FDMs. In this paper, a fourth-order FDM is optimized to improve prediction accuracy by considering both the helix angle and multi-mode. Based on the numerical stability computation results, it can firstly be observed that the stability lobe diagram (SLD) obtained using the proposed FDM has advantages over the existing SDM in the aspects of both the convergence rate and computational efficiency, and the effect of helix angle on the SLD is determined by both the number of teeth, radial immersion ratio and helix angle of the tool. Furthermore, a detailed investigation on how to detect chatter in milling experiments is made using an integrated method with time-frequency analysis. Finally, comparing the prediction result with the experimental milling results, it can be concluded that the proposed SLD with first three modes excels in prediction accuracy. A quantitative evaluation about the improvement of prediction accuracy is also made. The intention of proposing the updated FDM is to make the chatter prediction more accurate and efficient. Highlights • The proposed FDM considers the helix angle and multi-mode effects. • FDM with 4th-order interpolation for DDEs' state term is utilized for optimization. • An integrated method with time-frequency analysis is proposed to detect chatter. • Improved precision in chatter stability prediction in milling is demonstrated. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
29. Superconvergence analysis of a two-grid method for semilinear parabolic equations.
- Author
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Shi, Dongyang, Mu, Pengcong, and Yang, Huaijun
- Subjects
- *
DEGENERATE parabolic equations , *STOCHASTIC convergence , *SEMILINEAR elliptic equations , *INTERPOLATION , *NUMERICAL analysis - Abstract
Abstract In this paper, the superconvergence analysis of a two-grid method (TGM) is established for the semilinear parabolic equations. Based on the combination of the interpolation and Ritz projection technique, an important ingredient in the method, the superclose estimates in the H 1 -norm are deduced for the backward Euler fully-discrete TGM scheme. Moreover, through the interpolated postprocessing approach, the corresponding global superconvergence result is derived. Finally, some numerical results are provided to confirm the theoretical analysis, and also show that the computing cost of the proposed TGM is only half of the conventional Galerkin finite element methods (FEMs). [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
30. A multiobjective hybrid bat algorithm for combined economic/emission dispatch.
- Author
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Liang, Huijun, Liu, Yungang, Li, Fengzhong, and Shen, Yanjun
- Subjects
- *
MULTIPLE criteria decision making , *PROBABILITY theory , *ELECTRIC power systems , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
In this paper, a multiobjective hybrid bat algorithm is proposed to solve the combined economic/emission dispatch problem with power flow constraints. In the proposed algorithm, an elitist nondominated sorting method and a modified crowding-distance sorting method are introduced to acquire an evenly distributed Pareto Optimal Front. A modified comprehensive learning strategy is used to enhance the learning ability of population. Through this way, each individual can learn not only from all individual best solutions but also from the global best solutions (nondominated solutions). A random black hole model is introduced to ensure that each dimension in current solution can be updated individually with a predefined probability. This is not only meaningful in enhancing the global search ability and accelerating convergence speed, but particularly key to deal with high dimensional systems, especially large-scale power systems. In addition, chaotic map is integrated to increase the diversity of population and avoid premature convergence. Finally, numerical examples on the IEEE 30-bus, 118-bus and 300-bus systems, are provided to demonstrate the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
31. Numerical solutions for solving time fractional Fokker–Planck equations based on spectral collocation methods.
- Author
-
Yang, Yin, Huang, Yunqing, and Zhou, Yong
- Subjects
- *
NUMERICAL analysis , *FOKKER-Planck equation , *DISCRETIZATION methods , *JACOBI polynomials , *INTERPOLATION , *STOCHASTIC convergence - Abstract
In this paper, we consider the numerical solution of the time fractional Fokker-Planck equation. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Generalized two-step Maruyama methods for stochastic differential equations.
- Author
-
Ren, Quanwei and Tian, Hongjiong
- Subjects
- *
STOCHASTIC differential equations , *NUMERICAL analysis , *MEAN square algorithms , *STOCHASTIC convergence , *LINEAR statistical models - Abstract
In this paper, we propose generalized two-step Maruyama methods for solving Itô stochastic differential equations. Numerical analysis concerning consistency, convergence and numerical stability in the mean-square sense is presented. We derive sufficient and necessary conditions for linear mean-square stability of the generalized two-step Maruyama methods. We compare the stability region of the generalized two-step Maruyama methods of Adams type with that of the corresponding two-step Maruyama methods of Adams type and show that our proposed methods have better linear mean-square stability. A numerical example is given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
33. Verification and validation of numerical modelling of DTMB 5415 roll decay.
- Author
-
Mancini, Simone, Begovic, Ermina, Day, Alexander H., and Incecik, Atilla
- Subjects
- *
STOCHASTIC convergence , *COMPUTATIONAL fluid dynamics , *NUMERICAL analysis - Abstract
The paper presents a numerical roll damping assessment of the intact DTMB 5415 naval ship at zero speed. Free model motions from four experimental roll decays with initial heeling angle of 4.0, 13.5, 19.58 and 24.50 deg, performed previously at the University of Strathclyde, Glasgow, have been analysed and the one with 19.58 deg initial heeling has been chosen for the Computational Fluid Dynamic (CFD) analysis. All calculations are performed using CD Adapco Star CCM + software investigating the accuracy and efficiency of the numerical approach for case of high initial heeling angle of bare hull. In the numerical procedure the verification analysis of mesh refinement and time step was performed with the aim to investigate the numerical error/uncertainty. For grid refinement and time step, validation and verification procedure has been performed according to the Grid Convergence Index (GCI) method. Moreover, to verify the main source of the modelling error/uncertainty, the effect of degrees of freedom are evaluated, comparing the numerical results with the experimental results. Conclusions are identifying best practice for roll decay simulations commenting the accuracy of numerical results and required calculation time. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
34. The analysis of operator splitting methods for the Camassa–Holm equation.
- Author
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Zhan, Rui and Zhao, Jingjun
- Subjects
- *
STOCHASTIC convergence , *EQUATIONS , *NUMERICAL analysis , *LIE algebras , *LIPSCHITZ spaces - Abstract
In this paper, the convergence analysis of operator splitting methods for the Camassa–Holm equation is provided. The analysis is built upon the regularity of the Camassa–Holm equation and the divided equations. It is proved that the solution of the Camassa–Holm equation satisfies the locally Lipschitz condition in H 1 and H 2 norm, which ensures the regularity of the numerical solution. Through the calculus of Lie derivatives, we show that the Lie–Trotter and Strang splitting converge with the expected rate under suitable assumptions. Numerical experiments are presented to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
35. ADI schemes for valuing European options under the Bates model.
- Author
-
in 't Hout, Karel J. and Toivanen, Jari
- Subjects
- *
VON Neumann algebras , *STOCHASTIC convergence , *PARTIAL differential equations , *NUMERICAL analysis , *COMPUTER simulation - Abstract
This paper is concerned with the adaptation of alternating direction implicit (ADI) time discretization schemes for the numerical solution of partial integro-differential equations (PIDEs) with application to the Bates model in finance. Three different adaptations are formulated and their (von Neumann) stability is analyzed. Ample numerical experiments are provided for the Bates PIDE, illustrating the actual stability and convergence behaviour of the three adaptations. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
36. Three-steps modified Levenberg–Marquardt method with a new line search for systems of nonlinear equations.
- Author
-
Amini, Keyvan and Rostami, Faramarz
- Subjects
- *
NONLINEAR equations , *MATHEMATICAL bounds , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *MONOTONE operators , *NUMERICAL analysis - Abstract
Three steps modified Levenberg–Marquardt method for nonlinear equations was introduced by Yang (2013). This method uses the addition of the Levenberg–Marquardt (LM) step and two approximate LM steps as the trial step at every iteration. Using trust region technique, the global and biquadratic convergence of the method is proved by Yang. The main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique. Since the search direction of Yang method may be not a descent direction, standard line searches cannot be used directly. In this paper we propose a new nonmonotone third order Armijo type line search technique which guarantees the global convergence of this method while we use an adaptive LM parameter. It is proved that the convergence order of the new method is biquadratic. Numerical results show the new algorithm is efficient and promising. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
37. Reduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications.
- Author
-
Zhang, Huamin
- Subjects
- *
GENERALIZATION , *SYLVESTER matrix equations , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis - Abstract
In this paper, by constructing an objective function and using the gradient search, full-rank and reduced-rank gradient-based algorithms are suggested for solving generalized coupled Sylvester matrix equations. It is proved that the reduced-rank iterative algorithm is convergent for proper initial iterative values. By analyzing the spectral radius of the related matrices, the convergence properties are studied and the optimal convergence factor of the reduced-rank algorithm is determined. The relationship between the reduced-rank algorithm and the full-rank algorithm is discussed. Consequently, the computation load can be reduced greatly for solving a class of matrix equation. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
38. On parallel multisplitting block iterative methods for linear systems arising in the numerical solution of Euler equations.
- Author
-
Zhang, Cheng-yi, Luo, Shuanghua, and Xu, Zongben
- Subjects
- *
ITERATIVE methods (Mathematics) , *LINEAR systems , *NUMERICAL analysis , *EULER equations (Rigid dynamics) , *EXTRAPOLATION , *STOCHASTIC convergence - Abstract
The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As special cases the convergence of the parallel block generalized AOR (BGAOR), the parallel block AOR (BAOR), the parallel block generalized SOR (BGSOR), the parallel block SOR (BSOR), the extrapolated parallel BAOR and the extrapolated parallel BSOR methods are presented. Furthermore, the convergence of the parallel block iterative methods for linear systems with special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
39. Numerical stationary distribution and its convergence for nonlinear stochastic differential equations.
- Author
-
Liu, Wei and Mao, Xuerong
- Subjects
- *
NUMERICAL analysis , *DISTRIBUTION (Probability theory) , *STOCHASTIC convergence , *NONLINEAR equations , *STOCHASTIC differential equations - Abstract
To avoid finding the stationary distributions of stochastic differential equations by solving the nontrivial Kolmogorov–Fokker–Planck equations, the numerical stationary distributions are used as the approximations instead. This paper is devoted to approximate the stationary distribution of the underlying equation by the Backward Euler–Maruyama method. Currently existing results (Mao et al., 2005; Yuan et al., 2005; Yuan et al., 2004) are extended in this paper to cover larger range of nonlinear SDEs when the linear growth condition on the drift coefficient is violated. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
40. Steady-state response of thermoelastic half-plane with voids subjected to a surface harmonic force and a thermal source.
- Author
-
Zhu, Y.Y., Li, Y., and Cheng, C.J.
- Subjects
- *
THERMOELASTICITY , *ELECTRICAL harmonics , *STEADY-state responses , *NUMERICAL analysis , *STOCHASTIC convergence - Abstract
In this paper, the steady-state response for a thermoelastic half-plane with voids is studied, in which, the surface of the half-plane is partly subjected to a surface harmonic force and a thermal source. The semi-analytical solutions and the numerical solutions of the half-plane problems with three different materials are obtained from a semi-analytical method and a developed differential quadrature element method in this paper, respectively. The corresponding numerical results are compared and the effects of parameters are considered as well. It can be seen that the semi-analytical solutions and corresponding numerical solutions coincide with each other. This means that the differential quadrature element method is a very efficient method for seeking the numerical solutions of the half-plane problems with discontinuity, and it has some advantageous properties, such as small computational amount, high accuracy, and better convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
41. Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay.
- Author
-
Milošević, Marija
- Subjects
- *
NUMERICAL analysis , *NONLINEAR systems , *STOCHASTIC differential equations , *TIME delay systems , *EULER method , *STOCHASTIC convergence - Abstract
This paper represents the continuation of the analysis from papers Milošević (2011) [10] and Milošević (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
42. A spectral method for triangular prism.
- Author
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Li, Jingliang, Ma, Heping, Qin, Yonghui, and Zhang, Shuaiyin
- Subjects
- *
APPROXIMATION theory , *INTEGRALS , *STOCHASTIC convergence , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
In this paper, we study a spectral method for the triangular prism. We construct an approximation space in the “pole” condition in which the integral singularity is removed in a simple and effective way. We build a quasi-interpolation operator in the approximation space, and analyze its L 2 -error. Based on the quasi-interpolation, a triangular prism spectral method for the elliptic modal problem is studied. Furthermore, we extend this triangular prism spectral method to a triangular prism spectral element method. For the elliptic modal problem, we present the spectral element scheme and analyze the convergence. At last, we do some experiments to test the effectiveness of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
43. A fully distributed ADMM-based dispatch approach for virtual power plant problems.
- Author
-
Chen, Guo and Li, Jueyou
- Subjects
- *
NUMERICAL analysis , *MATHEMATICAL optimization , *STOCHASTIC convergence , *ELECTRIC inverters , *POWER distribution networks - Abstract
In this paper, a fully distributed approach is proposed for a class of virtual power plant (VPP) problems. By characterizing two specific VPP problems, we first give a comprehensive VPP formulation that maximizes the economic benefit subjected to the power balance constraint, line transmission limits and local constraints of all distributed energy resources (DERs). Then, utilizing the alternating direction method of multipliers and consensus optimization, a distributed VPP dispatch algorithm is developed for the general VPP problem. In particular, Theorem 1 is derived to show the convergence of the algorithm. The proposed algorithm is completely distributed without requiring a centralized controller, and each DER is regarded as an agent by implementing local computation and only communicates information with its neighbors to cooperatively find the globally optimal solution. The algorithm brings some advantages, such as the privacy protection and more scalability than centralized control methods. Furthermore, a new variant of the algorithm is presented for improving the convergence rate. Finally, several case studies are used to illustrate the efficiency and effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
44. On the convergence rate of Clenshaw–Curtis quadrature for integrals with algebraic endpoint singularities.
- Author
-
Wang, Haiyong
- Subjects
- *
STOCHASTIC convergence , *QUADRATURE domains , *INTEGRALS , *MATHEMATICAL singularities , *NUMERICAL analysis - Abstract
In this paper, we are concerned with Clenshaw–Curtis quadrature for integrals with algebraic endpoint singularities. An asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the Chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of Chebyshev polynomials. Numerical examples are provided to confirm our analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
45. A uniformly almost second order convergent numerical method for singularly perturbed delay differential equations.
- Author
-
Erdogan, Fevzi and Cen, Zhongdi
- Subjects
- *
STOCHASTIC convergence , *MATHEMATICAL singularities , *NUMERICAL solutions to delay differential equations , *NUMERICAL analysis , *FINITE difference method - Abstract
The purpose of this paper is to present a uniform finite difference method for the numerical solution of a second order singularly perturbed delay differential equation. The problem is solved by using a hybrid difference scheme on a Shishkin-type mesh. The method is shown to be uniformly convergent with respect to the perturbation parameter. Numerical experiments illustrate in practice the result of convergence proved theoretically. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
46. Improvement of third-order WENO-Z scheme at critical points.
- Author
-
Xu, Weizheng and Wu, Weiguo
- Subjects
- *
CRITICAL point (Thermodynamics) , *SMOOTHNESS of functions , *STOCHASTIC convergence , *TAYLOR'S series , *NUMERICAL analysis , *NONLINEAR analysis - Abstract
In this paper, we present an improved third-order WENO-Z scheme to improve the order of convergence at critical points. This scheme is constructed through slightly modifying the local smoothness indicators of the conventional WENO-Z scheme with the way of Taylor expansion for third-order convergence. The present scheme is proved to be close to third-order by several standard accuracy tests. The performance enhancement of the WENO scheme through this modification is verified on a variety of one-dimensional and two-dimensional standard numerical experiments. Numerical results indicate that the present scheme provides better results in comparison with the earlier third-order WENO schemes like WENO-JS3, WENO-Z3. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
47. An iteration method for solving the linear system [formula omitted].
- Author
-
Tian, Zhaolu, Tian, Maoyi, Zhang, Yan, and Wen, Pihua
- Subjects
- *
ITERATIVE methods (Mathematics) , *LINEAR systems , *STOCHASTIC convergence , *NUMERICAL analysis , *KRYLOV subspace - Abstract
In this paper, based on a convergence splitting of the matrix A , we present an inner–outer iteration method for solving the linear system A x = b . We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of A X B = C . Finally, several numerical examples are given to validate the performance of our proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
48. A new approach of superconvergence analysis for two-dimensional time fractional diffusion equation.
- Author
-
Shi, Dongyang and Yang, Huaijun
- Subjects
- *
DIFFERENTIAL equations , *FINITE element method , *INTERPOLATION , *NUMERICAL analysis , *STOCHASTIC convergence - Abstract
In this paper, a new approach of superconvergent estimate of bilinear finite element is established for two-dimensional time-fractional diffusion equation under fully-discrete scheme. The novelty of this approach is the combination technique of the interpolation and Ritz projection as well as the superclose estimate in H 1 -norm between them, which avoids the difficulty of constructing a postprocessing operator for Ritz projection operator, and reduces the regularity requirement of the exact solution. At the same time, three numerical examples are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
49. Highly efficient iterative algorithms for solving nonlinear systems with arbitrary order of convergence [formula omitted], [formula omitted].
- Author
-
Cordero, Alicia, Jordán, Cristina, Torregrosa, Juan R., and Sanabria-Codesal, Esther
- Subjects
- *
NONLINEAR systems , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *FINITE differences , *NUMERICAL analysis - Abstract
It is known that the concept of optimality is not defined for multidimensional iterative methods for solving nonlinear systems of equations. However, usually optimal fourth-order schemes (extended to the case of several variables) are employed as starting steps in order to design higher order methods for this kind of problems. In this paper, we use a non-optimal (in scalar case) iterative procedure that is specially efficient for solving nonlinear systems, as the initial steps of an eighth-order scheme that improves the computational efficiency indices of the existing methods, as far as the authors know. Moreover, the method can be modified by adding similar steps, increasing the order of convergence three times per step added. This kind of procedures can be used for solving big-sized problems, such as those obtained by applying finite differences for approximating the solution of diffusion problem, heat conduction equations, etc. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and Fisher’s equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the proposed schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
50. Adaptive techniques in SOLD methods.
- Author
-
Lukáš, Petr and Knobloch, Petr
- Subjects
- *
OSCILLATIONS , *STOCHASTIC convergence , *FINITE element method , *PIECEWISE constant approximation , *PIECEWISE linear approximation , *NUMERICAL analysis , *MATHEMATICAL optimization , *MATHEMATICAL models - Abstract
We present new results where free parameters in spurious oscillations at layers diminishing (SOLD) method are adaptively chosen. Provided numerical results are for conforming piecewise linear finite element space with free parameters from piecewise constant finite element space. We focus on a higher order convergence we discovered in previous paper by Lukáš (2015). [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
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