A semi-discretization method based on quartic splines for solving one-space-dimensional hyperbolic equations

In this paper, based on C³ quartic splines, a semi-discretization method containing two schemes is constructed to solve one-space-dimensional linear hyperbolic equations. It is shown that both schemes are unconditionally stable and their approximation orders are of O(k⁵+h⁴) and of O(k⁷+h⁴) with k and h being step sizes in time and space, respectively, which are much higher than those of other published schemes. A numerical example is presented and the results are compared with other published numerical results.     

Tension spline solution of nonlinear sine-Gordon equation

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders O(k²+k²h²+h²)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2}) and O(k²+k²h²+h⁴)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4}). At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

Further results on ultraconvergence derivative recovery for odd-order rectangular finite elements

For rectangular finite element, we give a superconvergence method by SPR technique based on the generalization of a new ultraconvergence record and the sharp Green function estimates, by which we prove that the derivative has ultra-convergence of order O(h ^k+3) (k ⩾ 3 being odd) and displacement has order of O(h ^k+4) (k ⩾ 4 being even) at the locally symmetry points.      

Correction of finite element estimates for Sturm-Liouville eigenvalues

Summary It is shown that a simple asymptotic correction technique of Paine, de Hoog and Anderssen reduces the error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by the finite element method, with linear hat functions and mesh lengthh, fromO(k ⁴ h ²) toO(k h ²). The result still holds when the matrix elements are evaluated by Simpson's rule, but if the trapezoidal rule is used the error isO(k ² h ²). Numerical results demonstrate the usefulness of the correction even for low values ofk.     

Compact difference schemes for heat equation with Neumann boundary conditions (II)

This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ² + h⁴) for interior mesh point approximation and O(τ² + h³) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ² + h⁴) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ² + h^3.5) while the numerical accuracy is O(τ² + h⁴), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ² + h^2.5), while the actual numerical accuracy is O(τ² + h³). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ² + h⁴) and O(τ² + h³), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ² + h⁴) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Correction of finite element eigenvalues for problems with natural or periodic boundary conditions

Recent results of Andrew and Paine for a regular Sturm-Liouville problem with essential boundary conditions are extended to problems with natural or periodic boundary conditions. These results show that a simple asymptotic correction technique of Paine, de Hoog and Anderssen reduces the error in the estimate of thekth eigenvalue obtained by the finite element method, with linear hat functions and mesh lengthh, fromO(k ⁴ h ²) toO(kh ²). Numerical results show the correction to be useful even for low values ofk.     

Spline methods for the solution of hyperbolic equation with variable coefficients

In this study, we developed the methods based on nonpolynomial cubic spline for numerical solution of second‐order nonhomogeneous hyperbolic partial differential equation. Using nonpolynomial cubic spline in space and finite difference in time directions, we obtained the implicit three level methods of O(k² + h²) and O(k² + h⁴). The proposed methods are applicable to the problems having singularity at x = 0, too. Stability analysis of the presented methods have been carried out. The presented methods are applied to the nonhomogeneous examples of different types. Numerical comparison with Mohanty's method (Mohanty, Appl Math Comput, 165 (2005), 229–236) shows the superiority of our presented schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

List multicoloring problems involving the k‐fold Hall numbers

We show that the four‐cycle has a k‐fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ?5k/3?; furthermore, the same is not true with shorter list lengths. In terms of h^(k)(G), the k ‐fold Hall number of a graph G, this result is stated as h^(k)(C₄)=2k??k/3?. For longer cycles it is known that h^(k)(C_n)=2k, for n odd, and 2k??k/(n?1)?≤h^(k)(C_n)≤2k, for n even. Here we show the lower bound for n even, and conjecture that this is the right value (just as for C₄). We prove that if G is the diamond (a four‐cycle with a diagonal), then h^(k)(G)=2k. Combining these results with those published earlier we obtain a characterization of graphs G with h^(k)(G)=k. As a tool in the proofs we obtain and apply an elementary generalization of the classical Hall–Rado–Halmos–Vaughan theorem on pairwise disjoint subset representatives with prescribed cardinalities. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 16–34, 2010.     

Quartic spline methods for solving one-dimensional telegraphic equations

In this paper, by using a quartic spline in space and generalized trapezoidal formula in time direction, one-dimensional telegraphic equations are solved. We present new classes of two-level schemes. The accuracy of these methods are O(k²+h⁴). It has been shown that by suitably choosing parameter, a high accuracy scheme of O(k³+h⁴) can be derived from our methods. Numerical results demonstrate the superiority of the new scheme.     

Finite Element Methods for Parabolic Stochastic PDE’s

We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank–Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h^1/2+k^1/4, which is known to be optimal. We also consider cases where k is much greater than h², and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank–Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method. Mathematics Subject Classifications (2000) 60H15, 60H35, 65N30, 35R60.     

A New Compact Finite Difference Method for Solving the Generalized Long Wave Equation

The aim of this article is to analyze a new compact finite difference method (CFDM) for solving the generalized regularized long wave (GRLW) equation. This method leads to a system of linear equations involving tridiagonal matrices and the rate of convergence of the method is of order O(k ² + h ⁴) where k and h are mesh sizes of time and space variables, respectively. Stability analysis of the method is investigated by the energy method and an error estimate is given. The propagation of single solitons and interaction of two solitary waves are applied to validate the method which is found to be accurate and efficient. Three invariants of the motion are evaluated to determine conservation properties of the method.     

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ² + h²) . For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ² + h²) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.     

On conditional edge-connectivity of graphs

1. IntroductionIn this paper, a graph G ~ (V,E) always means a simple graph (without loops andmultiple edges) with the vertex-set V and the edge-set E. We follow [1] for graph-theoreticalterllilnology and notation not defined here.It is well known that when the underlying topology of a computer interconnectionnetwork is modeled by a graph G, the edge-connectivity A(G) of G is an important measurefor fault-tolerance of the network. However, it has many deficiencies (see [2]). MotiVatedby t…     

Compact high-order finite-element method for elliptic transport problems with variable coefficients

We present a new conforming bilinear Petrov-Galerkin finite-element scheme for elliptic transport problems with variable coefficients. This scheme combines a generalized test function and artificial diffusion to achieve O(h⁴) grid-point accuracy on uniform stencils of 3 × 3 in two dimensions without resorting to the extended stencils of high-order elements. The method is compared with upwind and high-order finite-difference schemes and the standard Galerkin finite-element method for representative test problems. © 1994 John Wiley & Sons, Inc.     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

We present error estimates of a linear fully discrete scheme for a three-dimensional mass diffusion model for incompressible fluids (also called Kazhikhov–Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C ⁰-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete velocities, we first obtain point-wise stability estimates for the density, under the constraint lim_(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order O(h+k){\mathcal{O}(h+k)}) for regular enough solutions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).     

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H¹ error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ² + h⁴) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.     

RHPMDs and FHPMDs with k = 4 and Type 4 u

A (v, k, 1)-HPMD is called a frame (briefly, k-FHPMD), if the blocks of the HPMD can be partitioned into v partial parallel classes such that the complement of each partial parallel class is a group of the HPMD. A (v, k, 1)-HPMD is called resolvable (briefly, k-RHPMD), if the blocks of the HPMD can be partitioned into parallel classes. In this article, (i) we shall construct 3-FHPMDs of type 3⁶ and 21⁶ to completely settle the existence of 3-FHPMD of type h^u; (ii) we shall show that the necessary conditions for the existence of 4-FHPMD of type h^u are sufficient for the case h = 4; (iii) we shall show that the necessary conditions for the existence of 4-RHPMD of type h^u are sufficient for the case h = 4.