Table of Content

15 July 2018, Volume 36 Issue 4

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A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE LINEAR ELASTICITY PROBLEM IN MIXED FORM

Ruishu Wang, Ran Zhang

2018, 36(4):  469-491.  DOI: 10.4208/jcm.1701-m2016-0733

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In this paper, we use the weak Galerkin (WG) finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement field. For the WG methods, we define the weak function and the weak differential operator in an optimal polynomial approximation spaces. The optimal error estimates are given and numerical results are presented to demonstrate the efficiency and the accuracy of the weak Galerkin finite element method.

BLOCK-CENTERED FINITE DIFFERENCE METHODS FOR NON-FICKIAN FLOW IN POROUS MEDIA

Xiaoli Li, Hongxing Rui

2018, 36(4):  492-516.  DOI: 10.4208/jcm.1701-m2016-0628

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In this article, two block-centered finite difference schemes are introduced and analyzed to solve the parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. One scheme is Euler backward scheme with first order accuracy in time increment while the other is Crank-Nicolson scheme with second order accuracy in time increment. Stability analysis and second-order error estimates in spatial meshsize for both pressure and velocity in discrete L² norms are established on non-uniform rectangular grid. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.

HIGH ORDER STABLE MULTI-DOMAIN HYBRID RKDG AND WENO-FD METHODS

Fan Zhang, Tiegang Liu, Jian Cheng

2018, 36(4):  517-541.  DOI: 10.4208/jcm.1702-m2016-0707

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Recently, a kind of high order hybrid methods based on Runge-Kutta discontinuous Galerkin (RKDG) method and weighted essentially non-oscillatory finite difference (WENO-FD) scheme was proposed. Those methods are computationally efficient, however stable problems might sometimes be encountered in practical applications. In this work, we first analyze the linear stabilities of those methods based on the Heuristic theory. We find that the conservative hybrid method is linearly unstable if the numerical flux at the coupling interface is chosen to be ‘downstream’. Then we introduce two ways of healing this defect. One is to choose the numerical flux at the coupling interface to be ‘upstream’. The other is to employ a slope limiter function to enforce the hybrid method satisfying the local total variation diminishing (TVD) condition. In the end, numerical experiments are provided to validate the effectiveness of the proposed methods.

QUASI-NEWTON WAVEFORM RELAXATION BASED ON ENERGY METHOD

Yaolin Jiang, Zhen Miao

2018, 36(4):  542-562.  DOI: 10.4208/jcm.1702-m2016-0700

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A quasi-Newton waveform relaxation (WR) algorithm for semi-linear reaction-diffusion equations is presented at first in this paper. Using the idea of energy estimate, a general proof method for convergence of the continuous case and the discrete case of quasi-Newton WR is given, which appears to be the superlinear rate. The semi-linear wave equation and semi-linear coupled equations can similarly be solved by quasi-Newton WR algorithm and be proved as convergent with the energy inequalities. Finally several parallel numerical experiments are implemented to confirm the effectiveness of the above theories.

ANOMALOUS DIFFUSION IN FINITE LENGTH FINGERS COMB FRAME WITH THE EFFECTS OF TIME AND SPACE RIESZ FRACTIONAL CATTANEO-CHRISTOV FLUX AND POISEUILLE FLOW

Lin Liu, Liancun Zheng, Fawang Liu, Xinxin Zhang

2018, 36(4):  563-578.  DOI: 10.4208/jcm.1702-m2016-0627

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This paper presents an investigation on the anomalous diffusion in finite length fingers comb frame, the time and space Riesz fractional Cattaneo-Christov flux is introduced with the Oldroyds' upper convective derivative and the effect of Poiseuille flow is also taken into account. Formulated governing equation possesses the coexisting characteristics of parabolicity and hyperbolicity. Numerical solution is obtained by the L¹-scheme and shifted Grünwald formulae, which is verified by introducing a source item to construct an exact solution. The effects, such as time and space fractional parameters, relaxation parameter and the ratio of the pressure gradient and viscosity coefficient, on the spatial and temporal evolution of particles distribution and dynamic characteristics are shown graphically and analyzed in detail.

A MODIFIED PRECONDITIONER FOR PARAMETERIZED INEXACT UZAWA METHOD FOR INDEFINITE SADDLE POINT PROBLEMS

Xinhui Shao, Chen Li, Tie Zhang, Changjun Li

2018, 36(4):  579-590.  DOI: 10.4208/jcm.1702-m2016-0665

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The preconditioner for parameterized inexact Uzawa methods have been used to solve some indefinite saddle point problems. Firstly, we modify the preconditioner by making it more generalized, then we use theoretical analyses to show that the iteration method converges under certain conditions. Moreover, we discuss the optimal parameter and matrices based on these conditions. Finally, we propose two improved methods. Numerical experiments are provided to show the effectiveness of the modified preconditioner. All methods have fantastic convergence rates by choosing the optimal parameter and matrices.

HIGH ORDER COMPACT MULTISYMPLECTIC SCHEME FOR COUPLED NONLINEAR SCHRÖDINGER-KDV EQUATIONS

Lan Wang, Yushun Wang

2018, 36(4):  591-604.  DOI: 10.4208/jcm.1702-m2016-0789

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In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.

A NEW BOUNDARY CONDITION FOR RATE-TYPE NON-NEWTONIAN DIFFUSIVE MODELS AND THE STABLE MAC SCHEME

Kun Li, Youngju Lee, Christina Starkey

2018, 36(4):  605-626.  DOI: 10.4208/jcm.1703-m2015-0359

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We present a new Dirichlet boundary condition for the rate-type non-Newtonian diffusive constitutive models. The newly proposed boundary condition is compared with two such well-known and popularly used boundary conditions as the pure Neumann condition[1] and the Dirichlet condition by Sureshkumar and Beris[2]. Our condition is demonstrated to be more stable and robust in a number of numerical test cases. A new Dirichlet boundary condition is implemented in the framework of the finite difference Marker and Cell (MAC) method. In this paper, we also present an energy-stable finite difference MAC scheme that preserves the positivity for the conformation tensor and show how the addition of the diffusion helps the energy-stability in a finite difference MAC scheme-setting.