A method for improving the performance of the WENO5 scheme near discontinuities

WENO5 uses a convex combination of the polynomials reconstructed on the three stencils of ENO3 in order to achieve higher accuracy on smooth profiles. However, in some cases WENO5 generates oscillations or smears near discontinuities due to the time scheme used. Here, we present a method to reduce those oscillations without damping and this yields a sharper approximation. Our technique uses smoothness indicators to identify severe shocks and switches from WENO5 to ENO3. Numerical tests show that the behaviour of WENO5 is improved near discontinuities while preserving high accuracy on smooth profiles.     

Stability analysis of a difference scheme for the Navier-Stokes equations

An application of stability analysis by the energy method is made to a practical problem of unsteady viscous flow solved numerically by Fromm. The practical stability criterion for the scheme is determined and a rigorous proof of convergence to a smooth solution is given. It is also shown how to construct an energy for any 3-level scalar difference equation.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

Detailed error analysis for a fractional Adams method with graded meshes

We consider a fractional Adams method for solving the nonlinear fractional differential equation \(\,^{C}_{0}D^{\alpha }_{t} y(t) = f(t, y(t)), \, \alpha >0\), equipped with the initial conditions \(y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots , \lceil \alpha \rceil -1\). Here, α may be an arbitrary positive number and ?α? denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption \(\,^{C}_{0}D^{\alpha }_{t} y \in C^{2}[0, T]\), Diethelm et al. (Numer. Algor. 36, 31–52, 2004) introduced a fractional Adams method with the uniform meshes t _n = T(n/N),n = 0,1,2,…,N and proved that this method has the optimal convergence order uniformly in t _n , that is O(N ^?2) if α > 1 and O(N ^?1?α ) if α ≤ 1. They also showed that if \(\,^{C}_{0}D^{\alpha }_{t} y(t) \notin C^{2}[0, T]\), the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C ^m [0,T] for some \(m \in \mathbb {N}\) and 0 < α < m, the Caputo fractional derivative \(\,^{C}_{0}D^{\alpha }_{t} y(t) \) takes the form “\(\,^{C}_{0}D^{\alpha }_{t} y(t) = c t^{\lceil \alpha \rceil -\alpha } + \text {smoother terms}\)” (Diethelm et al. Numer. Algor. 36, 31–52, 2004), which implies that \(\,^{C}_{0}D^{\alpha }_{t} y \) behaves as t ^?α??α which is not in C ²[0,T]. By using the graded meshes t _n = T(n/N) ^r ,n = 0,1,2,…,N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t _n even if \(\,^{C}_{0}D^{\alpha }_{t} y\) behaves as t ^σ ,0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

Stability and error analysis for spectral Galerkin method for the Navier–Stokes equations with L2 initial data

In this article we consider the spectral Galerkin method with the implicit/explicit Euler scheme for the two‐dimensional Navier–Stokes equations with the L² initial data. Due to the poor smoothness of the solution on [0,1), we use the the spectral Galerkin method based on high‐dimensional spectral space H_M and small time step Δt² on this interval. While on [1,∞), we use the spectral Galerkin method based on low‐dimensional spectral space H_m(m = O(M^1/2)) and large time step Δt. For the spectral Galerkin method, we provide the standard H²‐stability and the L²‐error analysis. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Stability analysis for a fully discrete spectral scheme for Boussinesq systems

In this paper we perform a stability analysis of a fully discrete numerical method for the solution of a family of Boussinesq systems, consisting of a Fourier collocation spectral method for the spatial discretization and a explicit fourth order Runge–Kutta (RK4) scheme for time integration. Our goal is to determine the influence of the parameters, associated to this family of systems, on the efficiency and accuracy of the numerical method. This analysis allows us to identify which regions in the parameter space are most appropriate for obtaining an efficient and accurate numerical solution. We show several numerical examples in order to validate the accuracy, stability and applicability of our MATLAB implementation of the numerical method.     

Stability and error analysis for a spectral Galerkin method for the Navier‐Stokes equations with H2 or H1 initial data

In this article we consider a spectral Galerkin method with a semi‐implicit Euler scheme for the two‐dimensional Navier‐Stokes equations with H² or H¹ initial data. The H²‐stability analysis of this spectral Galerkin method shows that for the smooth initial data the semi‐implicit Euler scheme admits a large time step. The L²‐error analysis of the spectral Galerkin method shows that for the smoother initial data the numerical solution u exhibits faster convergence on the time interval [0, 1] and retains the same convergence rate on the time interval [1, ∞). © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Central limit theorem for Maxwellian molecules and truncation of the wild expansion

We prove an L¹ bound on the error made when the Wild summation for solutions of the Boltzmann equation for a gas of Maxwellian molecules is truncated at the n^th stage. This gives quantitative control over the only constructive method known for solving the Boltzmann equation. As such, it has recently been applied to numerical computation but without control on the approximation made in truncation. We also show that our bound is qualitatively sharp and that it leads to a simple proof of the exponentially fast rate of relaxation to equilibrium for Maxwellian molecules along lines originally suggested by McKean. © 2000 John Wiley & Sons, Inc.     

New strong convergence theorem of the inertial projection and contraction method for variational inequality problems

Numerical Algorithms - In this paper, we introduce a new algorithm which combines the inertial projection and contraction method and the viscosity method for solving monotone variational inequality...     

Note on error analysis for the spectral synthesis method with interfaces

Summary In a recent paper Meyer and Nelson [1] obtained error estimates for the Galerkin type spectral synthesis method relative to the continuous-energy, continuous-space, time independent neutron diffusion equation. These estimates were obtained under a coercivity condition which is not easily verifiable. The aim of this note is to replace that condition by a readily verifiable one.

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Zusammenfassung In einer kürzlichen Publikation haben Meyer und Nelson [1] Fehlerabschätzungen für eine Galerkin-ähnliche Spektralsynthesemethode, angewandt auf die zeitunabhängige Neutronen-diffusionsgleichung mit kontinuierlicher Abhängigkeit von Energie und Ort, diskutiert. Die Abschätzungen wurden unter einer nicht leicht überprüfbaren Zwangsbedingung erhalten. Das Ziel dieser Arbeit ist, diese Bedingungen durch eine bequem verifizierbare zu ersetzen.

     

An efficient spectral method and rigorous error analysis based on dimension reduction scheme for fourth order problems

In this paper, we propose an effective spectral method based on dimension reduction scheme for fourth order problems in polar geometric domains. First, the original problem is decomposed into a series of one‐dimensional fourth order problems by polar coordinate transformation and the orthogonal properties of Fourier basis function. Then the weak form and the corresponding discrete scheme of each one‐dimensional fourth order problem are derived by introducing polar conditions and appropriate weighted Sobolev spaces. In addition, we define the projection operators in the weighted Sobolev space and give its approximation properties, and further prove the error estimation of each one‐dimensional fourth order problem. Finally, we provide some numerical examples, and the numerical results show the effectiveness of our algorithm and the correctness of the theoretical results.     

A conservative SPH scheme using exact projection with semi-analytical boundary method for free-surface flows

This paper presents a novel SPH scheme for modelling incompressible and divergence-free flow with a free surface (IDFSPH) associated with semi-analytical wall boundary conditions. In line with the projection method, the velocity field is decoupled from the pressure field in the momentum equation. A Poisson equation, serving as the pressure solver, is obtained by which pressure field is decoupled completely from the velocity field. In particular, an exact projection scheme is deployed to fulfil the requirement of the divergence-free velocity field. The condition of incompressibility is satisfied by iteratively updating the density field till the convergence. The two-equation k–ε model is employed to describe the turbulence effects in Newtonian flows. It is shown that the discretised SPH schemes have the feature of both linear and angular momentum conservations. The semi-analytical wall method implements the appropriate integrals to evaluate the boundary contributions to the mass and momentum equations. In comparison to the boundary particle methods, it can greatly enhance the feasibility and efficiency with the complex geometries. The algorithm presented within this paper is applied to several academic test cases for which either analytical results or simulations with other methods are available. The comparisons verify that this scheme is provided with convincing efficiency and extensive applicability.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

On a joint strong approximation theorem for record and inter-record times

Summary We present a simple joint strong approximation for the logarithms of record and inter-record times from an exchangeable sequence, including an exact estimation for the rate of convergence in terms of upper and lower class functions of a Wiener process. The approach chosen here allows for simple proofs of exact and asymptotic (joint) results for record and inter-record times, such as the Law of Large Numbers (LLN), Central Limit Theorem (CLT) and Law of the Iterated Logarithm (LIL), and others.Research supported by the Air Force Office of Scientific Research Contract No. F49620 85 C0144     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014