Laguerre spectral approximation of Stokes’ first problem for third-grade fluid

A Laguerre–Galerkin method is proposed and analyzed for Quasilinear parabolic differential equation which arises from Stokes’ first problem for a third-grade fluid on a semi-infinite interval. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations to the transformed equations is developed and implemented. Effects of non-Newtonian parameters on the flow phenomena are analyzed and documented.     

Controllability method for acoustic scattering with spectral elements

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.     

A trapping principle and convergence result for finite element approximate solutions of steady reaction/diffusion systems

We consider nonlinear elliptic systems, with mixed boundary conditions, on a convex polyhedral domain Ω ⊂ R ^N . These are nonlinear divergence form generalizations of Δu = f(·, u), where f is outward pointing on the trapping region boundary. The motivation is that of applications to steady-state reaction/diffusion systems. Also included are reaction/diffusion/convection systems which satisfy the Einstein relations, for which the Cole-Hopf transformation is possible. For maximum generality, the theory is not tied to any specific application. We are able to demonstrate a trapping principle for the piecewise linear Galerkin approximation, defined via a lumped integration hypothesis on integrals involving f, by use of variational inequalities. Results of this type have previously been obtained for parabolic systems by Estep, Larson, and Williams, and for nonlinear elliptic equations by Karátson and Korotov. Recent minimum and maximum principles have been obtained by Jüngel and Unterreiter for nonlinear elliptic equations. We make use of special properties of the element stiffness matrices, induced by a geometric constraint upon the simplicial decomposition. This constraint is known as the non-obtuseness condition. It states that the inward normals, associated with an arbitrary pair of an element’s faces, determine an angle with nonpositive cosine. Drăgănescu, Dupont, and Scott have constructed an example for which the discrete maximum principle fails if this condition is omitted. We also assume vertex communication in each element in the form of an irreducibility hypothesis on the off-diagonal elements of the stiffness matrix. There is a companion convergence result, which yields an existence theorem for the solution. This entails a consistency hypothesis for interpolation on the boundary, and depends on the Tabata construction of simple function approximation, based on barycentric regions. This work was supported by the National Science Foundation under grant DMS-0311263.     

Projektive Newton-Verfahren und Anwendungen auf nichtlineare Randwertaufgaben

Summary In this paper we consider the following Newton-like methods for the solution of nonlinear equations. In each step of the Newton method the linear equations are solved approximatively by a projection method. We call this a Projective Newton method. For a fixed projection method the approximations often are the same as those of the Newton method applied to a nonlinear projection method. But the efficiency can be increased by adapting the accuracy of the projection method to the convergence of the approximations. We investigate the convergence and the order of convergence for these methods. The results are applied to some Projective Newton methods for nonlinear two point boundary value problems. Some numerical results indicate the efficiency of these methods.

     

Numerical solution of the Poisson equation over hypercubes using reduced Chebyshev polynomial bases

We describe how to use new reduced size polynomial approximations for the numerical solution of the Poisson equation over hypercubes. Our method is based on a non-standard Galerkin method which allows test functions which do not verify the boundary conditions. Numerical examples are given in dimensions up to 8 on solutions with different smoothness using the same approximation basis for both situations. A special attention is paid on conditioning problems.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-minimization methods for Hamilton–Jacobi equations: the one-dimensional case

A new approximation technique based on L ¹-minimization is introduced. It is proven that the approximate solution converges to the viscosity solution in the case of one-dimensional stationary Hamilton–Jacobi equation with convex Hamiltonian. This material is based upon work supported by the National Science Foundation grant DMS-0510650. J.-L. Guermond is on leave from LIMSI, UPRR 3251 CNRS, BP 133, 91403 Orsay Cedex, France.     

Some current questions on solving linear elliptic problems

Summary Common properties of elliptic problems and their solutions are compared with those of difference equations, with emphasis on the classical Dirichlet problem. Resistance network analogies are used to construct discrete harmonic functions; a new 9 POINT NET approximation is defined on elementary domains whose solutions haveO(h ⁴) max-norm error. The convection-diffusion equation is studied with especial emphasis on new fast iterative methods for solving its difference approximations. A new analysis of the structure of the matrix representation of difference approximations reveals special features which can naturally be associated with diffusion and with convective processes.Dedicated to R. S. Varga on the occasion of his sixtieth birthday     

Two-grid discretization schemes for nonlinear Schrödinger equations

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.     

Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

A conservative pressure-correction scheme for transient simulations of reacting flows

An algorithm is presented for numerical simulations of time-dependent low Mach number variable density flows with an arbitrary amount of scalar transport equations and a complex equation of state. The pressure-correction type algorithm is based on a segregated solution formalism. It is conservative and guarantees stable results, regardless of the difference in density between neighboring cells. Furthermore, states are predicted which exactly match the equation of state. In the one-dimensional example, considering non-premixed flames, a simplified flamesheet model is used to describe the combustion of fuel and oxidizer. We demonstrate that the predicted states exactly correspond to the equation of state. We illustrate the accuracy improvement due to higher order formulation and demonstrate grid convergence.     

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form under the very general structural condition . To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P). Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000     

Error analysis and applications of the Fourier-Galerkin Runge-Kutta schemes for high-order stiff PDEs

An integrating factor mixed with Runge-Kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. In this paper, the novel hybrid Fourier-Galerkin Runge-Kutta scheme, with the aid of an integrating factor, is proposed to solve nonlinear high-order stiff PDEs. Error analysis and properties of the scheme are provided. Application to the approximate solution of the nonlinear stiff Korteweg-de Vries (the 3rd order PDE, dispersive equation), Kuramoto-Sivashinsky (the 4th order PDE, dissipative equation) and Kawahara (the 5th order PDE) equations are presented. Comparisons are made between this proposed scheme and the competing method given by Kassam and Trefethen. It is found that for KdV, KS and Kawahara equations, the proposed method is the best.     

A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks

The numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered. Since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods, based on global interpolation by analytic functions, show low accuracy. In order to circumvent this issue, a meshfree modification of the method of fundamental solutions is developed, where the approximation basis is enriched by an extra span of corner adapted non-smooth shape functions. The high accuracy of the new method is illustrated by solving several boundary value problems for the Helmholtz equation, modelling physical phenomena from the fields of room acoustics and acoustic resonance.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

On richardson extrapolation for finite difference methods on regular grids

Summary Difference solutions of partial differential equations can in certain cases be expanded by even powers of a discretization parameterh. If we haven solutions corresponding to different mesh widthsh ₁,...,h _n we can improve the accuracy by Richardson extrapolation and get a solution of order 2n, yet only on the intersection of all grids used, i.e. normally on the coarsest grid. To interpolate this high order solution with the same accuracy in points not belonging to all grids, we need 2n points in an interval of length (2n–1)h ₁.This drawback can be avoided by combining such an interpolation with the extrapolation byh. In this case the approximation depends only on grid points in an interval of length 3/2h ₁. The length of this interval is independent of the desired order.By combining this approach with the method of Kreiss, boundary conditions on curved boundaries can be discretized with a high order even on coarse grids.This paper is based on a lecture held at the 5th Sanmarinian University Session of the International Academy of Sciences San Marino, at San Marino, 1988-08-27-1988-09-05     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.