Differential-equation-based representation of truncation errors for accurate numerical simulation

High-order compact finite difference schemes for two-dimensional convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h⁴) schemes conform to a 3 × 3 stencil. We show that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h²) centred differencing and commonly used O(h) and O(h³) upwinding schemes.     

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h^1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ω^γ) for γ?1, then the optimized asymptotic convergence factor is 1 – O(ω^(1–2γ)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

New numerical method for partial differential equations. 1: Application to the diffusion equation

In this paper a new, highly accurate method called PH is presented for the numerical integration of partial differential equations. The method is applied for the solution of the one-dimensional diffusion equation. Upon integrating the equation within a subdomain of space and time using the prismoidal approximation, a three-point implicit scheme is obtained with a truncation error of order O(k⁴, h⁶), where k and h represent the time and space steps respectively. The method is stable under the condition s = αk/h² ? S(δ), where the function S(δ) increases as the parameter δ decreases from 1/12 to negative values. In practice the method behaves as unconditionally stable upon choosing an appropriate value for δ. A new formula is also adopted for the implementation of a Neumann boundary condition, introducing a truncation error of order O(h⁴). Numerical solutions are obtained incorporating Dirichlet and Neumann boundary conditions. The results prove that our method is far more accurate than any other-implicit or explicit method.     

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h ³) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h ³-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h ⁵) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.     

Mixed boundary value problem of heat conduction for infinite slab

Summary An axisymmetric steady state heat conduction boundary value problem having mixed boundary conditions on both faces of an infinite slab, is reduced to a pair of Fredholm integral equations of the second kind. For large values of h, the slab thickness, a solution correct to O(h ^–6) is obtained by expanding the kernels in power series.Presently at Imperial College, London.     

Bifurcations of a Thin Plate-Strip Excited Transversally and Axially

The complex vibrations and bifurcations of plates modeled as systemswith infinite degrees-of-freedom are considered. Both theBubnov–Galerkin with high-order approximations and finite differencemethods with approximation O(h ⁴)are applied. In addition, the calculation ofthe Lyapunov exponents of the system is performed, and the results arecompared to those derived by Bennetin's method. Some examples of newnonlinear phenomena exhibited by the considered systems are reported.     

Improving the accuracy of the least-squares finite element approximation of the linearized steady Euler equations by an embedding method

The standard least-squares finite element method for the linearized Euler equations turns out to be inaccurate. This method is studied in detail for a system of composite type, obtained by transformation of the linearized Euler equations. The shortcomings of the method are clarified and an embedding method is constructed. It is shown numerically that this new method is O(h²)-accurate.     

Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²?h³)_x = ??·(h³?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²?h³)_x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h³?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.     

On the equation of deflection of a thick plate

Summary Approximate equations of the deflection of a thick plate are derived from fundamental equations of a three-dimensional elastic body, by expanding components of displacement into power series in platethickness h and then by truncating at appropriate terms of 0(h ⁿ ).The method proposed here enables us to give a systematic treatment to obtain approximate equations with any desired accuracy in the sense of increasing order of h. As an example, a thick plate is treated under distributed pressure acting at its upper surface.

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Über die Gleichung zur Durchbiegung einer dicken Platte

Übersicht Näherungsgleichungen für die Durchbiegung einer dicken Platte werden aus den drei-dimensionalen Fundamentalgleichungen der Elastizität hergeleitet, wobei die Verschiebungen erst nach Potenzreihen der Plattendicke h entwickelt und danach an passenden Termen von 0(h ⁿ ) abgebrochen werden.Die hier präsentierte Methode ermöglicht uns ein systematisches Verfahren, um die Näherungsgleichungen mit einer beliebig gewünschten Genauigkeit im Sinne zunehmender Ordnung von h zu gewinnen. Als ein Beispiel wird eine dicke Platte unter verteilter Belastung an ihrer Deckfläche behandelt.

     

Symmetries of degenerate center singularities of plane vector fields

Let D² ì \mathbbR² {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ì \mathbbR² O \subset {\mathbb{R}^2} and let F be a smooth vector field such that O is the unique singular point of F, and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically, O is a “center” singularity. Let D⁺ (F) {\mathcal{D}^{+} }(F) be the group of all diffeomorphisms of D ² that preserve the orientation and orbits of F. Recently, the author described the homotopy type of D⁺ (F) {\mathcal{D}^{+} }(F) under the assumption that the 1-jet j ¹ F(O) of F at O is nondegenerate. In this paper, the degenerate case j ¹ F(O) is considered. Under additional “nondegeneracy assumptions” on F, the path components of D⁺ (F) {\mathcal{D}^{+} }(F) with respect to distinct weak topologies are described. These conditions imply that, for each h ? D⁺ (F) h \in {\mathcal{D}^{+} }(F) , its path component in D⁺ (F) {\mathcal{D}^{+} }(F) is uniquely determined by the 1-jet of h at O.     

Squeeze flow of a power-law fluid between non-parallel plates

Squeeze flow in the gap between non-parallel circular plates of radius R is discussed. The test material is assumed to be a power-law fluid, with a no-slip boundary condition at the plates. If the mean separation between the plates is h, and the angle of inclination between the plates is ? ? h/R, the force on the plates is perturbed only at O(?²) and is increased by less than 10% if ? < 0.35h/R. A torque O(?) tends to return the plates to a parallel configuration.     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone??s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h ³). Using h ³?Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h ⁵), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.     

Stationary Flows of Shear Thickening Fluids in 2D

We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \mathbbR²{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions h having the property that h′ (t)/t increases (shear thickening case).     

Long waves on a layer of a visco-elastic fluid down an inclined plane

Summary In this paper the finite amplitude stability of long waves on a layer of a second-order fluid flowing down an inclined plane is discussed. A systematic expansion procedure in terms of a parameterµ, which is the ratio of the undisturbed layer thickness to a representative length down the plane, is developed and solutions are obtained toO(µ ³). It is found that weakly non-linear monochromatic waves tend to attain equilibrium states for Weber numbers ofO(µ ^–2). This equilibrium amplitude first increases with increase in the elastic parameterM, reaches a maximum and then decreases withM. It is also shown that the second fluid behaves like a Newtonian fluid with its viscosity reduced through division by the factor 1 + (5M/2).

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Zusammenfassung In dieser Veröffentlichung wird die Stabilität von langen Wellen mit endlicher Amplitude in der Schicht einer Flüssigkeit zweiter Ordnung diskutiert, die längs einer geneigten Ebene abfließt. Es wird ein systematisches Entwicklungsverfahren nach einem Parameterµ angegeben, der das Verhältnis der ungestörten Schichtdicke zu einer repräsentativen Länge längs der Ebene beschreibt, und es werden Lösungen bis zur OrdnungO(µ ³) erhalten. Man findet, daß schwach nicht-lineare monochromatische Wellen für Weber-Zahlen der OrdnungO(µ ^–2) einem Gleichgewichtszustand zustreben. Die Gleichgewichtsamplitude nimmt mit wachsendem elastischem ParameterM zuerst zu, erreicht ein Maximum und fällt dann mitM wieder ab. Es wird schließlich noch gezeigt, daß sich die Flüssigkeit zweiter Ordnung wie eine newtonsche Flüssigkeit verhält, deren Viskosität jedoch durch Division durch einen Faktor 1 + (5M/2) reduziert ist.

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Superconvergence analysis of bi-<Emphasis Type="Italic">k</Emphasis>-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(h^k+1) in the H¹ norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^k+1 + τ²) in the H¹ norm can be obtained in the Crank-Nicolson fully discrete scheme.     

Simulation of shockwave propagation with a thermal lattice Boltzmann model

A two‐dimensional 19‐velocity (D2Q19) lattice Boltzmann model which satisfies the conservation laws governing the macroscopic and microscopic mass, momentum and energy with local equilibrium distribution order O(u⁴) rather than the usual O(u³) has been developed. This model is applied to simulate the reflection of shockwaves on the surface of a triangular obstacle. Good qualitative agreement between the numerical predictions and experimental measurements is obtained. As the model contains the higher‐order terms in the local equilibrium distribution, it performs much better in terms of numerical accuracy and stability than the earlier 13‐velocity models with the local equilibrium distribution accurate only up to the second order in the velocity u. Copyright © 2003 John Wiley & Sons, Ltd.     

Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet

We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form

Dynamics of inner mappings

We study properties of invariant sets of dynamical systems generated by inner mappings. We prove that if x is a nonwandering point of a finitely multiple inner mapping, then not only its positive trajectory O⁺(x) consists of nonwandering points but also the negative trajectory O⁻(x) contains at least one partial semitrajectory consisting of nonwandering points.     

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

A new type of Galerkin finite element for first-order initial-value problems(IVPs) is proposed. Both the trial and test functions employ the same m-degreed polynomials. The adjoint equation is used to eliminate one degree of freedom(DOF) from the test function, and then the so-called condensed test function and its consequent condensed Galerkin element are constructed. It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of...