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SIAM J. on Numerical Analysis

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1992

Volume 29, Issue 6, pp. 1505-1820


The Convergence Rate of Approximate Solutions for Nonlinear Scalar Conservation Laws

Haim Nessyahu and Eitan Tadmor

SIAM J. Numer. Anal. 29, pp. 1505-1519 (15 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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Let $\{ {v^\varepsilon (x,t)} \}_{\varepsilon > 0} $ be a family of approximate solutions for the nonlinear scalar conservation law $u_t + f(u)_x = 0$ with $C_0^1 $-initial data. Assume that $\{ {v^\varepsilon (x,t)} \}$ are ${\textit{Lip}}^ + $-stable in the sense that they satisfy Oleinik’s E-entropy condition. It is shown that if these approximate solutions are ${\textit{Lip}}' $-consistent, i.e., if $\| {v^\varepsilon ( \cdot ,0) - u( \cdot ,0)} \|_{{\textit{Lip}}'(x)} + \| {v_t^\varepsilon + f(v^\varepsilon )_x } \|_{{\textit{Lip}}'(x,t)} = \mathcal {O}(\varepsilon )$, then they converge to the entropy solution, and the convergence rate estimate $\| {v^\varepsilon ( \cdot ,t) - u( \cdot ,t)} \|_{{\textit{Lip'}}(x)} = \mathcal {O}(\varepsilon )$ holds. Consequently, the familiar $L^p $-type and new pointwise error estimates are derived.
These convergence rate results are demonstrated in the context of entropy satisfying finite-difference and Glimm’s schemes.

Convergence of Spectral Methods for Burgers’ Equation

Weinan E

SIAM J. Numer. Anal. 29, pp. 1520-1541 (22 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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In this paper, a general framework is presented for analyzing numerical methods for the evolutionary equations that admit semigroup formulations. This framework is then applied to spectral and pseudospectral methods for the Burgers’ equation, using trigonometric, Chebyshev, and Legendre polynomials. Optimal order of convergence is obtained, which implies the spectral accuracy of these methods.

Optimum Positive Linear Schemes for Advection in Two and Three Dimensions

P. L. Roe and D. Sidilkover

SIAM J. Numer. Anal. 29, pp. 1542-1568 (27 pages) | Cited 29 times

Online Publication Date: July 31, 2006

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In this paper the optimal linear, positive schemes for constant-coefficient advection in two or three dimensions are presented. These are the generalizations of first-order upwinding in one dimension. By comparison with a dimension-by-dimension treatment the optimum schemes have much lower numerical diffusion, and permit larger timesteps.

On Maximum Norm Convergence of Multigrid Methods for Two-Point Boundary Value Problems

Arnold Reusken

SIAM J. Numer. Anal. 29, pp. 1569-1578 (10 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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Multigrid methods applied to standard linear finite element discretizations of linear elliptic two-point boundary value problems are considered. In the multigrid method damped Jacobi or damped Gauss–Seidel is used as a smoother. It is shown that the contraction number with respect to the maximum norm has an upper bound which is smaller than one and independent of the mesh size.

Analysis and Convergence of the MAC Scheme. I. The Linear Problem

R. A. Nicolaides

SIAM J. Numer. Anal. 29, pp. 1579-1591 (13 pages) | Cited 23 times

Online Publication Date: July 31, 2006

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The MAC (Marker and Cell) discretization of fluid flow is analysed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard ${\bf H}^1 $ and $L^2 $ norms are given for the velocity and pressure, respectively.

Viscous Shock Profiles and Primitive Formulations

S. Karni

SIAM J. Numer. Anal. 29, pp. 1592-1609 (18 pages) | Cited 12 times

Online Publication Date: July 31, 2006

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Weak solutions of hyperbolic systems in primitive (nonconservative) form for which a consistent conservation form exists are considered. It is shown that for primitive formulations, shock relations are not uniquely defined by the states to either side of the shock, but also depend on the viscous path connecting the two. Consistent viscous shock profiles are enforced by adding scheme-dependent small viscous perturbations that account for leading order conservation errors. The resulting primitive algorithm is conservative to the order of the approximation. One-dimensional Euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms. While not eliminating conservation errors entirely, it is found that for a wide range of problems, both conservative and primitive flow calculations are of comparable quality.

Hierarchical Conforming Finite Element Methods for the Biharmonic Equation

P. Oswald

SIAM J. Numer. Anal. 29, pp. 1610-1625 (16 pages) | Cited 16 times

Online Publication Date: July 31, 2006

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The paper deals with hierarchical bases in spaces of conforming $C^1 $ elements in connection with the approximate solution of the biharmonic equation \[ \Delta ^2 u = f\quad {\text{in }}\Omega ,\qquad u = \frac{{\partial u}}{{\partial n}} = 0\quad {\text{on }}\partial \Omega \]
on a plane polygonal domain $\Omega $. Two different composite finite elements are studied: piecewise quadratic Powell–Sabin elements and piecewise cubic elements of Clough–Tocher type.
The main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization (the so-called hierarchical multilevel method) will yield suboptimal convergence rates in comparison with standard multigrid schemes.

The Pseudospectral Method for Third-Order Differential Equations

Weizhang Huang and David M. Sloan

SIAM J. Numer. Anal. 29, pp. 1626-1647 (22 pages) | Cited 11 times

Online Publication Date: July 31, 2006

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Generalized quadrature rules are derived which assist in the selection of collocation points for the pseudospectral solution of differential equations. In particular, it is shown that for an $n$th-order differential equation in one space dimension with two-point derivative boundary conditions, an ideal choice of interior collocation points is the set of zeros of a Jacobi polynomial. The pseudospectral solution of a third-order initial-boundary value problem is considered and accuracy is assessed by examining how well the discrete eigenproblem approximates the continuous one. Convergence is established for a special choice of collocation points and numerical results are included to demonstrate the viability of the approach.

Noniterative Approximations to the Solution of the Matrix Riccati Differential Equation

M. C. Delfour and A. Ouansafi

SIAM J. Numer. Anal. 29, pp. 1648-1693 (46 pages)

Online Publication Date: July 31, 2006

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The object of this paper is to present new approximation schemes for the nonlinear matrix Riccati differential equation. They are obtained from any one-step or multistep method applied to the original linear quadratic control problem. They lead to the same type of schemes. However only one matrix inversion is required at each discretization node even if the Riccati equation is a nonlinear equation. This important computational advantage is obtained without altering the original nodal asymptotic convergence rate. It is proved that this rate is the same as the one of the initially chosen scheme.

Differential/Algebraic Equations As Stiff Ordinary Differential Equations

Michael Knorrenschild

SIAM J. Numer. Anal. 29, pp. 1694-1715 (22 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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This paper deals with the relation between differential/algebraic equations (DAEs) and certain stiff ODEs and their respective discretizations by implicit Runge–Kutta methods. For that purpose for any DAE a singular perturbed ODE is constructed such that the DAE is its reduced problem and the solution of the ODE converges in some sense to that of the DAE. Thus the DAE can be interpreted as an infinitely stiff ODE. An analysis of the discretization error of this singular perturbed system gives insight into the relationship of order-reduction phenomena observed for stiff ODEs to that for DAEs. Analysis of a general class of singularly perturbed problems and their discretizations is not attempted; however, the technique of treating singularly perturbed problems and DAEs in a unified way is new and can possibly be applied to other systems and their discretizations as well. Since asymptotic expansions are not used, but an approach similar to the ones used in B-convergence theory is applied, one can derive error bounds that are uniform in the perturbation parameter as well as in the stepsize and do not suffer from restrictions on the ratio of these parameters. This enables one to relate the order of convergence achieved for DAEs to the order of B-convergence. This phenomenon is discussed for several classes of Runge–Kutta methods and illustrated with a numerical example.

On the Representation of Operators in Bases of Compactly Supported Wavelets

G. Beylkin

SIAM J. Numer. Anal. 29, pp. 1716-1740 (25 pages) | Cited 102 times

Online Publication Date: July 31, 2006

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This paper describes exact and explicit representations of the differential operators, ${{d^n } / {dx^n }}$, $n = 1,2, \cdots $, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.
Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring $O(N\log N)$ operations for computing the wavelet coefficients of all $N$ circulant shifts of a vector of the length $N = 2^n $ is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141–183]) may be reduced from $O(N)$ to $O(\log ^2 N)$ significant entries.

Numerical Solution of a Functional Equation on a Circle

Anne C. Morlet and Jens Lorenz

SIAM J. Numer. Anal. 29, pp. 1741-1768 (28 pages)

Online Publication Date: July 31, 2006

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In this article, two numerical methods are analyzed and applied to compute an invariant closed curve for a particular class of maps: a second-order scheme based on linear interpolation and a scheme based on interpolation by cubic splines. In suitable coordinates, the invariant curve is determined by the solution of a functional equation on a circle. A linearized version of this functional equation is studied in detail. Stability of the schemes is discussed and error estimates are derived depending on the smoothness of the solution. Roughly speaking, the analysis predicts superior behavior of the spline scheme when the solution is sufficiently smooth, but a more reliable behavior of the lower-order scheme in situations when a smooth solution cannot be expected. This is confirmed in numerical examples.
The analysis of the linear functional equation does not readily generalize to nonlinear equations, but the algorithms for the linear equation can be used as a building block for a treatment of nonlinear equations. This is demonstrated in the last section where an invariant closed curve of the Poincare map is computed for the periodically forced van der Pol oscillator. The formal Newton iteration in function space is discretized by the spline interpolation scheme.

Fast Algorithms for Nonsmooth Compact Fixed-Point Problems

M. Heinkenschloss, C. T. Kelley, and H. T. Tran

SIAM J. Numer. Anal. 29, pp. 1769-1792 (24 pages) | Cited 15 times

Online Publication Date: July 31, 2006

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A fast algorithm for compact fixed-point problems with nonsmooth nonlinearities is designed and analyzed. The algorithm is a combination of an extension of the Atkinson–Brakhage–Nyström algorithm for smooth problems and a generalization of work by Yamamoto and Chen for nonsmooth problems. A critical structural hypothesis in the general theory is explicitly verified in the context of problems that can be expressed as integral equations with certain types of nonsmoothness. The work is motivated by problems in combat modeling. In particular, we consider the solution of an optimality system that arises in control of competitive systems.

Reduced SQP Methods for Parameter Identification Problems

K. Kunisch and E. W. Sachs

SIAM J. Numer. Anal. 29, pp. 1793-1820 (28 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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Parameter estimation problems are formulated as constrained, regularized optimization problems. Reduced SQP methods with BFGS update are analyzed to solve these infinite-dimensional optimization problems. Rate of convergence results are given and numerical feasibility of the resulting algorithms is demonstrated.
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