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A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are (O(tau ^2 + h^2)) for the velocity in the (L^infty )-norm and the surface elevation in the (L^2)-norm, where (tau ) denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.

Abstract

In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples.

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in (L^2(I;L^2(Omega )^d)) and (L^2(I;H^1(Omega )^d)) norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type (L^2(I;H^1(Omega ))) error estimate are new even for scalar parabolic problems.

Abstract

A convection–diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Durán type meshes, as well as on coarser Durán type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments.

The aim of this paper is twofold. First, we prove (L^p) estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter h. Actually, we show that at the singularity the discrete Green’s function is of order (h^{-1}), which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.

In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, ((p=4,5,6,7)) and (A(alpha ))-stable for order p, ((p=8,9,ldots , 14).) Compared to the existing BDF based method, i.e. class (2+1,) hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF.

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order (alpha in (1,2).) The error has the asymptotic expansion ( big ( d_{3} tau ^{3- alpha } + d_{4} tau ^{4-alpha } + d_{5} tau ^{5-alpha } + cdots big ) + big ( d_{2}^{*} tau ^{4} + d_{3}^{*} tau ^{6} + d_{4}^{*} tau ^{8} + cdots big ) ) at any fixed time (t_{N}= T, N in {mathbb {Z}}^{+}), where (d_{i}, i=3, 4,ldots ) and (d_{i}^{*}, i=2, 3,ldots ) denote some suitable constants and (tau = T/N) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order (alpha in (1,2)) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

We propose two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral (SPEWIFP) methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. In this regime, the solution of Dirac equation is highly oscillatory in time because of the small parameter (0 <varepsilon ll 1) which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition (tau lesssim 1) and preserve the modified energy and modified mass in the discrete level. Although our methods can only preserve the modified energy and modified mass instead of the original energy and mass, our methods are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are often implicit. Through rigorous error analysis, we give the error bounds of the methods at (O(h^{m_0} + tau ^2/varepsilon ^2)) where h is mesh size, (tau ) is time step and the integer (m_0) is determined by the regularity conditions. These error bounds indicate that, to obtain the correct numerical solution in the simultaneously massless and nonrelativistic regime, our methods request the (varepsilon )-scalability as (h = O(1)) and (tau = O(varepsilon )) which is better than the (varepsilon )-scalability of the finite difference (FD) methods: (h =O(varepsilon ^{1/2})) and (tau = O(varepsilon ^{3/2})). Numerical experiments confirm that the theoretical results in this paper are correct.

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings.

In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal (H^1) error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions.