• Abstract

We present an approximate algorithm to solve only one variable out of a linear system defined by a matrix with off-diagonal exponential decay entries (including the practically most important class of band limited matrices) via a sub-linear system. This approach thus enables us to solve any subset of solution variables. Parallel implementation of such approximate schemes for every variable enables us to solve the linear system with computational time independent of the matrix size.

• Abstract

In this paper, two types of higher-order linearized multistep finite difference schemes are proposed to solve non-Fickian delay reaction-diffusion equations. For the first scheme, the equations are discretized based on the backward differentiation formulas in time and compact finite difference approximations in space. The global convergence of the scheme is proved rigorously with convergence order O(τ^2 + h^4) in the maximum norm. Next, a linearized noncompact multistep finite difference scheme is presented and the corresponding error estimate is established. Finally, extensive numerical examples are carried out to demonstrate the accuracy and efficiency of the schemes, and some comparisons with the implicit Euler scheme in the literature are presented to show the effectiveness of our schemes.

• Abstract

We consider a nonlinear variational wave equation that models the dynamics of the director field in nematic liquid crystals with high molecular rotational inertia. Being derived from an energy principle, energy stability is an intrinsic property of solutions to this model. For the twodimensional case, we design numerical schemes based on the discontinuous Galerkin framework that either conserve or dissipate a discrete version of the energy. Extensive numerical experiments are performed verifying the scheme's energy stability, order of convergence and computational efficiency. The numerical solutions are compared to those of a simpler first-order Hamiltonian scheme. We provide numerical evidence that solutions of the 2D variational wave equation loose regularity in finite time. After that occurs, dissipative and conservative schemes appear to converge to different solutions.

• Abstract

This paper analyzes an existing 4-node hybrid mixed-shear-projected quadrilateral element MiSP4, presented by Ayad, Dhatt and Batoz (Int. J. Numer. Meth. Engng 1998, 42: 1149- 1179) for Reissner-Mindlin plates, which behaves robustly in numerical benchmark tests. This method is based on Hellinger-Reissner variational principle, where continuous piecewise isoparametric bilinear interpolations, as well as the mixed shear interpolation ⁄ projection technique of MITC family, are used for the approximations of displacements, and piecewise-independent equilibrium modes are used for the approximations of bending moments ⁄ shear stresses. Due to local elimination of the parameters of moments ⁄ stresses, the computational cost of MiSP4 element is almost the same as that of the conforming bilinear quadrilateral displacement element. We show that the element is free from shear locking in the sense that the error bound in the derived a priori estimate is independent of the plate thickness.

• Abstract

In this paper, a numerical approach for the simulation of a dynamical model with damping defined by the Riemann-Liouville fractional derivative and with uncertainty, that is fuzziness, is discussed. The proposed method exploits differential quadrature rules and a Picard- like recursion. The convergence is formally discussed. Some example applications, in the linear and nonlinear regime, confirm the theoretical achievements.

• Abstract

In this paper, we proposed a regularization model based on second-order total variation for CT image reconstruction, which could eliminate the ‘staircase’ caused by total variation (TV) minimization. Moreover, some properties of second-order total variation were investigated, and a primal-dual algorithm for the proposed model was presented. Some numerical experiments for various projection data were conducted to demonstrate the efficiency of the proposed model and algorithm.

• Abstract

We introduce a flux recovery scheme for the computed solution of a quadratic immersed finite element method introduced by Lin et al. in [13]. The recovery is done at nodes and interface point first and by interpolation at the remaining points. In the case of piecewise constant diffusion coefficient, we show that the end nodes are superconvergence points for both the primary variable p and its flux u. Furthermore, in the case of piecewise constant diffusion coefficient without the absorption term the errors at end nodes and interface point in the approximation of u and p are zero. In the general case, flux error at end nodes and interface point is third order. Numerical results are provided to confirm the theory.

• Abstract

Yang's wavewise entropy inequality [19] is verified for β-schemes which, when m = 2 and under a mild technique condition, guarantees the convergence of the schemes to the entropy solutions of convex conservation laws in one-dimensional scalar case. These schemes, constructed by S. Osher and S. Chakravarthy [13], are based on unwinding principle and use E-schemes as building blocks with simple flux limiters, without which all of them are even linearly unstable. The total variation diminishing property of these methods was established in the original work of S. Osher and S. Chakravarthy.

• Abstract

In this paper, we discuss the backward Euler method along with its linearized version for the Kelvin-Voigt viscoelastic fluid flow model with non zero forcing function, which is either independent of time or in L^∞(L²). After deriving some bounds for the semidiscrete scheme, a priori estimates in Dirichlet norm for the fully discrete scheme are obtained, which are valid uniformly in time using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical result for sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are derived, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Even when f = 0, the present result improves upon earlier result of Bajpai et al. (IJNAM,10 (2013), pp.481-507) in the sense that error bounds in this article depend on 1 ⁄ \sqrt{K} as against 1 ⁄ K^r, r ≥ 1. Finally, numerical experiments are conducted which confirm our theoretical findings.