SIAM Digital Library
 
 
 

SIAM J. on Numerical Analysis

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

1982

Volume 19, Issue 6, pp. 1091-1304


Large Time Step Shock-Capturing Techniques for Scalar Conservation Laws

Randall J. Leveque

SIAM J. Numer. Anal. 19, pp. 1091-1109 (19 pages) | Cited 24 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
For a scalar conservation law $u_t = f(u)_x $ with $f''$ of constant sign, the first order upwind difference scheme is a special case of Godunov’s method. The method is equivalent to solving a sequence of Riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level. The difference scheme is stable (and the solutions to the associated sequence of Riemann problems do not interact) provided the Courant number $v$ is less than 1. By allowing and explicitly handling such interactions, it is possible to obtain a generalized method which is stable for $v$ much larger than 1. In many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods. In particular, shocks can be correctly computed with virtually no smearing. The generalized method is rather unorthodox and still has some problems associated with it. Nonetheless, preliminary results are quite encouraging.

Discrete Approximations of Cosine Operator Functions. I

Ronald H. W. Hoppe

SIAM J. Numer. Anal. 19, pp. 1110-1128 (19 pages) | Cited 1 time

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
In the present paper we are concerned with the approximation of cosine operator functions which appear in a natural way in the study of the Cauchy problem for second order evolution equations. We derive both qualitative and quantitative convergence theorems characterizing the convergence of cosine operator functions in terms of their infinitesimal generators, and we discuss the impact of these results with respect to the approximate solution of the corresponding Cauchy problems.

An Enthalpty Formulation of the Stefan Problem

R. E. White

SIAM J. Numer. Anal. 19, pp. 1129-1157 (29 pages) | Cited 11 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
In this paper the implicit time discretization of $E_t - \nabla \cdot K\nabla T = f$, the enthalpy formulation of the Stef an problem, is considered. This generates the algebraic system $E + A\beta (E) = \eta $, where $E$, $\beta (E)$, $\eta \in {}^ + \mathbb{R}^L $, $A$ is an $M$-matrix and $\beta (E)$ is the “inverse” of the enthalpy function. The algebraic equation is solved by a modification of the Gauss–Seidel method and convergence is proved. The existence and uniqueness of a weak solution with $T \in W_2^{1,0} (Q_T )$ is established by Rothe’s method. An application to thermal energy storage units which utilize phase change materials is given. Heat transfer via both diffusion and convection of the liquid phase change material is considered.

A Numerical Solution of the Enthalpy Formulation of the Stefan Problem

R. E. White

SIAM J. Numer. Anal. 19, pp. 1158-1172 (15 pages) | Cited 12 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
We give an algorithm for approximating the solution of the Stefan problem. This problem may have thermal properties and forcing terms that depend on the space variable, temperature or temperature gradients. The resulting nonlinear algebraic problem has the form $E + \Delta tA(E)\beta (E) = \eta (E)$, where $\beta (E) = (\beta _j (E_j ))$, $\eta (E) = (\eta _j (E_j ))$ and $A(E) = (a_{ij} (E))$ is an $M$-matrix for each $E \in {}^ + \mathbb{R}^L $, where $L =$ number of nodes. The concept of $A(E)$ being uniformly irreducibly diagonally dominate on ${}^ + \mathbb{R}^L $ is introduced and studied. These results are used to establish convergence of a sequence, given by the algorithm, to a solution of the above algebraic system.

Block Iterative Methods for Elliptic and Parabolic Difference Equations

Seymour V. Parter and Michael Steuerwalt

SIAM J. Numer. Anal. 19, pp. 1173-1195 (23 pages) | Cited 7 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
Direct iterative methods for solving the linear system $AX = Y$ split $A$ into a difference $M - N$. By viewing $N$ as a weak multiplication operator, we determine the convergence rates of block direct iterative methods for elliptic and parabolic difference equations. The difference equations may arise from very general partial differential equations on general domains in $m$ space dimensions.

Numerical Approximation of a Periodic Linear Parabolic Problem

Christine Bernardi

SIAM J. Numer. Anal. 19, pp. 1196-1207 (12 pages) | Cited 1 time

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
We approximate a periodic linear parabolic problem using a Galerkin method for the space variable and a spectral method for the time variable. Some optimal error estimates are derived.

On a Finite Element Method to Solve the Criticality Eigenvalue Problem for the Transport Equation

Jean Descloux and Mitchell Luskin

SIAM J. Numer. Anal. 19, pp. 1208-1219 (12 pages)

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
A finite element discretization of the criticality eigenvalue problem for a one-dimensional model of the transport equation is analyzed. The existence of spurious eigenvalues for this procedure is demonstrated. However, it is shown that all of the spurious eigenvalues are larger than the smallest correct eigenvalue. This gives a justification for the use of the inverse power method to solve the discretized problem for the criticality eigenvalue, which is the smallest eigenvalue.

On the Solution of Block Tridiagonal Systems of Linear Algebraic Equations Having a Special Structure

J. R. Cash

SIAM J. Numer. Anal. 19, pp. 1220-1232 (13 pages) | Cited 1 time

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
Recently the present author has developed some high-order finite difference formulae for the approximate numerical integration of general two-point boundary value problems for ordinary differential equations. The algebraic equations arising from using these formulae in conjunction with a modified Newton iteration scheme are block tridiagonal with an additional special structure. Efficient algorithms for solving these equations are given and these appear to make high order finite difference schemes particularly attractive for the numerical solution of general two-point boundary value problems.

A Fast Solver Free of Fill-In for Finite Element Problems

M. R. Li, B. Nour-Omid, and B. N. Parlett

SIAM J. Numer. Anal. 19, pp. 1233-1242 (10 pages) | Cited 5 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
A new algorithm for solving finite element problems is presented. It blends a preconditioned conjugate gradient iteration into a direct factorization method. The goal was to reduce fill to a negligible level and thus reduce storage requirements, but the algorithm turned out to be faster than its rivals for an important class of problems.

A Trace Minimization Algorithm for the Generalized Eigenvalue Problem

Ahmed H. Sameh and John A. Wisniewski

SIAM J. Numer. Anal. 19, pp. 1243-1259 (17 pages) | Cited 27 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
An algorithm for computing a few of the smallest (or largest) eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem $Ax = \lambda Bx$ is presented. The matrices $A$ and $B$ are assumed to be symmetric, and haphazardly sparse, with $B$ being positive definite. The problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace. The rate of convergence of the method is established, and a technique for improving it is discussed. Numerical experiments and comparisons with other methods are presented.

Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex

Michael G. Duff

SIAM J. Numer. Anal. 19, pp. 1260-1262 (3 pages) | Cited 51 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
A simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex.

Nonpolynomial and Inverse Interpolation for Line Search: Synthesis and Convergence Rates

J. Barzilai and A. Ben-Tal

SIAM J. Numer. Anal. 19, pp. 1263-1277 (15 pages) | Cited 2 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
The rate of convergence of line search algorithms based on general interpolating functions is derived and is shown to be independent of the particular interpolating function used. This result holds for the root finding problem $f(x) = 0$ as well. We show how inverse interpolation can be used in conjunction with the line search problem and derive its rate of convergence. Our analysis suggests that one-point line search algorithms (in particular Newton's method) are inefficient in a sense. Two-point algorithms using rational interpolating functions are recommended.

Monotone and Convex Approximation by Splines: Error Estimates and a Curve Fitting Algorithm

R. K. Beatson

SIAM J. Numer. Anal. 19, pp. 1278-1285 (8 pages) | Cited 6 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
We obtain Jackson type estimates for the approximation of increasing or convex functions by splines with the same property. These estimates are new when the knots are not uniformly spaced. The method of proof motivates an algorithm for fitting a curve to data with local convexity preservation. The properties of this algorithm are developed and some examples are given.

A Fast Algorithm for Smoothing Data on a Rectangular Grid while Using Spline Functions

Paul Dierckx

SIAM J. Numer. Anal. 19, pp. 1286-1304 (19 pages) | Cited 7 times

Online Publication Date: July 17, 2006

Full Text: | Download PDF

Show Abstract
An efficient computational method is presented for fitting a bivariate spline function to a set of measured data on a rectangular grid. The coefficients in the $B$-spline representation of this spline are obtained by the solution of a linear system which can be arranged in a matrix form, conformable withthe Kronecker product of two band matrices of small size and bandwidth. The number of knots of the spline and their positions are determined automatically. Instead the algorithm expects a parameter to control the tradeoff between closeness of fit and smoothness of fit.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Entropy and Its Applications
  2. A Precision Approximation of the Gamma Function
  3. An Efficient Factorization for the Group Inverse
  4. Simplified Reliabilities for Consecutive-$k$-out-of-$n$ Systems
  5. On Iterative Solution for Linear Complementarity Problem with an $H_{+}$-Matrix
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close