Showing papers in "SIAM Journal on Numerical Analysis in 1985"
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TL;DR: In this paper, error estimates for fully discretized two-and three-dimensional vortex methods are given and a new way of evaluating the stretching of vorticity in 3D vortex methods is introduced.
Abstract: We give error estimates for fully discretized two- and three-dimensional vortex methods and introduce a new wy of evaluating the stretching of vorticity in three-dimensional vortex methods. The convergence theory of Beale and Majda is discussed and a simple proof of Cottet’s consistency result is presented. We also describe how to obtain accurate two-dimensional vortex methods in which the initial computational points are distributed on the nodes of nonrectangular grids, and compare several three-dimensional vortex methods.
357 citations
TL;DR: In this article, the fundamental solutions method for boundary value problems for elliptic homogeneous equations was proposed. But the fundamental solution method is not suitable for the case of the Laplacian.
Abstract: We consider a procedure for solving boundary value problems for elliptic homogeneous equations, known as the fundamental solutions method. We prove its applicability for some second order operators as well as for fourth order ones. The boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest. In a specific case of the Laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions. Numerical examples are given, among which are simple Signorini’s problems for harmonic and Lame operators.
341 citations
TL;DR: An almost uniform triangulation of the two-sphere, derived from the icosahedron, is presented, and a procedure for discretization of a partial differential equation using this triangular grid is described.
Abstract: We present an almost uniform triangulation of the two-sphere, derived from the icosahedron, and describe a procedure for discretization of a partial differential equation using this triangular grid. The accuracy of our procedure is described by a strong theoretical estimate, and verified by large-scale numerical experiments. We also describe a data structure for this spherical discretization that allows fast computation on either a vector computer or an asynchronous parallel computer.
268 citations
TL;DR: Time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time-truncation errors than those of standard methods.
Abstract: Miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations. The pressure equation is elliptic, while the concentration equation is parabolic but normally convection-dominated. A sequential backward-difference time-stepping scheme is defined; it approximates the pressure by a standard Galerkin procedure and the concentration by a combination of a Galerkin method and the method of characteristics. Optimal convergence rates in $L^2 $ and $H^1 $ are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations. Time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time-truncation errors than those of standard methods. Numerical results published elsewhere have confirmed that larger time steps are appropriate...
244 citations
TL;DR: In this article, the Raviart-Thomas mixed method was used for the model problem and the solution was obtained from the solution of the nonconforming Galerkin method modified in a virtually cost free manner.
Abstract: For the model problem $ - {\operatorname{div}}(a
abla u) = f$, we show how the approximate solution produced by the Raviart–Thomas [Lecture Notes in Mathematics 606, Springer-Verlag, Berlin, 1977] mixed method (of lowest degree) can be obtained from the solution of the $P_1 $ nonconforming Galerkin method modified in a virtually cost-free manner.
205 citations
TL;DR: Semidiscrete generalizations of the second order extension of Godunov’s scheme, known as the MUSCL scheme, are constructed, starting with any three point “E” scheme, used to approximate scalar conservation laws in one space dimension.
Abstract: Semidiscrete generalizations of the second order extension of Godunov’s scheme, known as the MUSCL scheme, are constructed, starting with any three point “E” scheme. They are used to approximate scalar conservation laws in one space dimension. For convex conservation laws, each member of a wide class is proven to be a convergent approximation to the correct physical solution. Comparison with another class of high resolution convergent schemes is made.
179 citations
TL;DR: This paper proves the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations in terms of the generalized condition number $\kappa $ of the matrix A and the smoothing matrix B.
Abstract: In this paper, we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations. Our theory is presented entirely in terms of the generalized condition number $\kappa $ of the matrix A and the smoothing matrix B. This leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations; the properties of the elliptic equation and discretization procedure enter only when we seek to estimate $\kappa $, just as in the case of most standard iterative methods. Here we consider the fundamental two-level iteration, and the V and W cycles of the j-level iteration ($j > 2$). We prove that the V and W cycles converge even when only one smoothing iteration is used. We present several examples of the computation of $\kappa $ using both Fourier analysis and standard finite element techniques. We compare the predictions of our theorems with the actual rate of convergence. Our analysis also shows that...
156 citations
TL;DR: For the nonlinear eigenvalue problem, the residual inverse iteration with variable shift is defined in this article, where the convergence rate is at least linear with convergence factor proportional to the variance of the residual.
Abstract: For the nonlinear eigenvalue problem $A(\hat \lambda )\hat x = 0$, where $A( \cdot )$ is a matrix-valued operator, residual inverse iteration with shift $\sigma $ is defined by \[ a^{(l + 1)} : = {\text{const. }}(x^{(l)} - A(\sigma )^{ - 1} A(\lambda _{l + 1} )x^{(l)} ),\] where $\lambda _{l + 1} $ is an appropriate approximation of $\hat \lambda $. In the linear case, $A(\lambda ) = A - \lambda I$, this is theoretically equivalent to ordinary inverse iteration, but the residual formulation results in a considerably higher limit accuracy when the residual $A(\lambda _{l + 1} )x^{(l)} = Ax^{(l)} - \lambda _{l + 1} x^{(l)} $ is accumulated in double precision. In the nonlinear case, if $\sigma $ is sufficiently close to $\hat \lambda $, convergence is at least linear with convergence factor proportional to $| {\sigma - \hat \lambda } |$. As with ordinary inverse iteration, the convergence can be accelerated by using variable shifts.
149 citations
TL;DR: A new trust region strategy for equality constrained minimization is developed and global as well as local superlinear convergence theorems are proved for various versions.
Abstract: In unconstrained minimization, trust region algorithms use directions that are a combination of the quasi-Newton direction and the steepest descent direction, depending on the fit between the quadratic approximation of the function and the function itself.Algorithms for nonlinear constrained minimization problems usually determine a quasi-Newton direction and use a line search technique to determine the step. Since trust region strategies have proved to be successful in unconstrained minimization, we develop a new trust region strategy for equality constrained minimization. This algorithm is analyzed and global as well as local superlinear convergence theorems are proved for various versions.We demonstrate how to implement this algorithm in a numerically stable way. A computer program based on this algorithm has performed very satisfactorily on test problems; numerical results are provided.
140 citations
TL;DR: A way to approximate the solution between mesh points is proposed which is applicable to some important formulas and theoretical support is much better than that of interpolation in the popular variable order, variable step Adams codes.
Abstract: Runge–Kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations. In contrast to the Adams methods, there is no natural way to approximate the solution between mesh points. A way to accomplish this is proposed which is applicable to some important formulas. Its theoretical support is much better than that of interpolation in the popular variable order, variable step Adams codes.
116 citations
TL;DR: In this article, the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two-point boundary-value problems by finite difference or finite element methods is considered.
Abstract: We consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two-point boundary-value problems by finite difference or finite element methods. An analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary-value problems in one dimension. Error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria.
TL;DR: In this paper, an interval enclosure for the set of slopes of an arithmetic expression where x ranges over an interval X is given, and an O(n)-time algorithm is given to compute the interval enclosure.
Abstract: For an arithmetic expression $f(x)$ involving N rational operations, an $O(N)$ algorithm is given which computes an interval enclosure for the set of slopes $f[ {x,z} ]$ where x ranges over an interval X. Applications to real and complex centered forms are given, resulting in improvements over previous results by Ratschek [SIAM J. Numer. Anal., 17 (1980), pp. 656–662] and Petkovic [Freiburger Intervall-Berichte, 83 (2) (1983), pp. 33–50].
TL;DR: In this paper, the authors considered the problem of finding paths of turning points in solutions of nonlinear systems having two parameters, where the paths are solutions of a particular extended system of non-linear equations.
Abstract: This paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters. It is well known that these paths are solutions of a particular extended system of nonlinear equations. In this paper both regular points and simple turning points in the extended system are related to the local geometry of the solution surface of the original nonlinear system. A description is given of numerical methods both for solving the extended system and for calculating certain quantities which determine the local geometry of the solution surface. Applications to perturbed bifurcation, to the formation of isolas, and to the calculation of the multiplicity of solutions are also discussed. Numerical examples are given.
TL;DR: In this article, a generalization of the Godunov-based method for solving conservation laws is proposed, which can be applied for arbitrarily large time steps and is easy to implement and may also be useful in other contexts, such as mesh refinement or shock tracking.
Abstract: A generalization of Godunov–s method for solving systems of conservation laws is proposed which can be applied for arbitrarily large time steps. Interactions of waves from neighboring Riemann problems are handled in an approximate but conservative manner that is exact for linear problems. For nonlinear systems it is found that better accuracy and sharper resolution of discontinuities is often obtained with Courant numbers somewhat larger than those allowed in Godunov's method. We explore the reasons for this behavior and, more generally, the effects of approximating wave interactions linearly. This linearization is easy to implement and may also be useful in other contexts, such as mesh refinement or shock tracking. A large time step generalization of the random choice method is also mentioned.
TL;DR: In this article, an extensive order analysis is given for the local and global errors of certain classes of Runge-Kutta methods applied to general nonlinear, stiff initial value problems.
Abstract: An extensive order analysis is given for the local and global errors of certain classes of Runge–Kutta methods applied to general nonlinear, stiff initial value problems. It turns out that there are three order levels: for linear problems $y' = Ay$ the classical orders apply, for general nonlinear problems, however, two types of order reductions occur. There a distinction has to be made between stiff problems where only $\| {f_y } \|$ is “large” and stiff problems where also some of the other derivatives $f_t ,f_{tt} ,f_{yy} , \cdots $, are “large.” For these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h. The order levels for both classes of problems are considerably lower than the respective conventional orders. These order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems.
TL;DR: In this article, a two-stage Runge-Kutta algorithm for vector Ito (and, by transform, also Stratonovich) stochastic differential equations with multiplicative noise has been developed.
Abstract: A two-stage, Runge–Kutta algorithm for vector Ito (and, by transform, also Stratonovich) stochastic differential equations with multiplicative noise has been developed. The method is second order accurate; but, for vanishing drift the algorithm yields a martingale independent of step size. Several examples are included to illustrate our method. A discussion of errors shows that sample size can be as important as truncation.
TL;DR: In this article, the authors proposed a method for which the stability limit is extended by treating the linear dispersive $u_{xxx} $ term implicitly, which can be implemented without solving linear systems by integrating in time in the Fourier space and discretizing the nonlinear $uu_x $ term by leap frog.
Abstract: A full leap frog Fourier method for integrating the Korteweg–de Vries (KdV) equation $u_t + uu_x - \varepsilon u_{xxx} = 0$ results in an $O(N^{ - 3} )$ stability constraint on the time step, where N is the number of Fourier modes used. This stability limit is much more restrictive than the accuracy limit for many applications.In this paper we propose a method for which the stability limit is extended by treating the linear dispersive $u_{xxx} $ term implicitly. Thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit. The implicit method is implemented without solving linear systems by integrating in time in the Fourier space and discretizing the nonlinear $uu_x $ term by leap frog. A second method we propose uses basis functions which solve the linear part of the KdV equation and leap frog for time integration. A linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that ...
TL;DR: An algorithm is presented to carry out the construction of a piecewise linear manifold along which H(x) = \theta <_\infty < \varepsilon $ in an efficient fashion.
Abstract: A simplicial method is used to approximate the solution manifold to a system of nonlinear equations, $H(x) = \theta $, where $H:\mathbb{R}^{N + K} \to \mathbb{R}^N $ The method begins at a point $x_0 $ in the solution set where the derivative $DH(x_0 )$ is of full rank. Given any $\varepsilon > 0$, a piecewise linear manifold is constructed along which $\| {H(x)} \|_\infty < \varepsilon $. An algorithm is presented to carry out this construction in an efficient fashion.
TL;DR: In this paper, an algorithm for interpolating monotone data, given on a rectangular grid, with a $C^1 $ quadratic spline surface was presented.
Abstract: $C^1 $ monotone quadratic splines are analyzed. This motivates an algorithm for interpolating monotone data, given on a rectangular grid, with a $C^1 $ monotone quadratic spline surface. Error estimates, an operation count and numerical examples are given.
TL;DR: Several classes of implicit Runge–Kutta methods are shown to be $BS$-stable: Gauss, Radau IA and Radau IIA schemes.
Abstract: New stability concepts—$BS$-stability and $BSI$-stability (internal $BS$-stability)—are introduced. $BS$-stability is a modification and extension of B-stability and enables the derivation of order results for Runge–Kutta methods applied to general nonlinear stiff initial value problems. These order results (B-consistency, B-convergence) are not affected by stiffness. Several classes of implicit Runge–Kutta methods are shown to be $BS$-stable: Gauss, Radau IA and Radau IIA schemes.
TL;DR: In this article, an algebraic convergence theory for a class of multigrid methods applied to positive definite self-adjoint linear operator equations is presented. But it does not consider the nonstationary and nonsymmetric relaxation schemes.
Abstract: In two earlier papers [SIAM J. Numer. Anal., 19 (1982), pp. 924–929; 21 (1984), pp. 255–262], we developed an algebraic convergence theory for a class of multigrid methods applied to positive definite self-adjoint linear operator equations. The purpose of the present paper is to extend these results by eliminating an earlier approximation order restriction, developing additional rate estimates and allowing for very general relaxation schemes, including those that are nonstationary and nonsymmetric. These results apply to most well-known iterative methods and preconditioners.
TL;DR: It is shown that the error in Godunov’s scheme is bounded by the same expression and this result is used to analyze a variant of a large time step numerical method of LeVeque that may be viewed as a combination ofGodunov's scheme and a method of Dafermos.
Abstract: The expected error in $L^1 (R)$ at time T for Glimm’s scheme when applied to a scalar conservation law is bounded by \[\left( {h + \frac{2}{{\sqrt 3 }}\left( {\frac{h}{{\Delta t}}} \right)^{{1 / 2}} } \right)\left\| {u_0 } \right\|_{BV({\bf R})} ,\] where h is the mesh size and $\Delta t$ is the time step. We show that the error in Godunov’s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of LeVeque that may be viewed as a combination of Godunov’s scheme and a method of Dafermos.
TL;DR: In this article, a spatial discretization of the Stokes problem in a domain of O(n 2 ) was studied, where the unknowns being the velocity and the pressure; optimal error was obtained by a finite element method.
Abstract: We study a spatial discretization of the Stokes problem in a domain of $\mathbb{R}^2 $ or $\mathbb{R}^3 $ by a finite element method, the unknowns being the velocity and the pressure; optimal error...
TL;DR: In this article, a simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y'(t) = py(t + qy(t - 8), where p and q are complex constants, and the delay 8 is a positive constant.
Abstract: A new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y'(t) = py(t) + qy(t - 8), where p and q are complex constants, and the delay 8 is a positive constant. The technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py(t) term with the region corresponding to the delay term qy(t - 8). A sufficient condition for stability is that these regions be disjoint. The technique is used to show that for each A-stable, A(a) -stable, or stiffly stable linear multistep formula for ordinary differential equations, there is a correspond- ing linear multistep formula for delay differential equations with analogous stability properties. The analogy does not extend, however, to A-stable one-step formulas.
TL;DR: The notion of (div)-semistability is developed in this article to study the behavior of the bilinear/constant and linear-constant "box" elements in detail.
Abstract: It is shown that i) the bilinear/constant and linear/constant “box” elements are not uniformly stable and ii) certain element pairs derived from those above by restricting the pressure spaces are stable. The notion of (div)-semistability is developed to study the behavior of the bilinear/constant and linear/constant “box” elements in detail.
TL;DR: In this paper, the well-conditioning of a boundary value problem is shown to be related to bounding two quantities, one involving the boundary conditions and the other involving the Green's function.
Abstract: The well-conditioning of a boundary value problem is shown to be related to bounding two quantities, one involving the boundary conditions and the other involving the Green’s function. In the case of a solution dichotomy, they are related to known stability results. These results easily explain why difficulties arise using superposition and reduced superposition. It is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem. Some factorization methods for the multiple shooting method are considered. These show the close relationships between multiple shooting, the stabilized march, and invariant imbedding, and one factorization leads to an efficient new way to implement each of these three methods. Many of the results apply for collocation and finite difference methods, too. Finally, it is shown, both theroetically and computationally, when matrix compactification can lead to difficulty.
TL;DR: In this paper, a least square formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods, and closely related arguments are shown to establish convergence estimates.
Abstract: A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.
TL;DR: In this article, it was shown that in the simplest case, this extra derivative is necessary and sufficient to achieve optimal order convergence for finite element methods for the wave equation, and it is well known that such an extra derivative of regularity is necessary.
Abstract: It is well known that finite element methods for the wave equation have optimal order convergence. However, known results require an apparent extra derivative of regularity in the solution to obtain this rate of convergence. We show that, in the simplest case, this extra derivative is necessary.
TL;DR: A variety of isospectral flows arise from Lax pairs and can all be interpreted in terms of the QR decomposition for nonsingular matrices, providing a new method of solving the eigenvalue problem.
Abstract: In this paper we consider a variety of isospectral flows on the set of $n \times n$ matrices. These flows arise from Lax pairs and can all be interpreted in terms of the $QR$ decomposition for nonsingular matrices. The asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem.
TL;DR: It is shown that the present method based on uniform mesh provides O(h^2 )-convergent approximations for all $\alpha \in (0,1)$ and reduces to the classical second order method for $y'' = f(x,y)$.
Abstract: A new finite difference method based on uniform mesh is given for the (weakly) singular two-point boundary value problem: $(x^\alpha y')' = f(x,y)$, $y(0) = A$, $y(1) = B$, $0 < \alpha < 1$. Under quite general conditions on $f'$ and $f''$, we show that our present method based on uniform mesh provides $O(h^2 )$-convergent approximations for all $\alpha \in (0,1)$. Our method is based on one evaluation of f and for $\alpha = 0$ it reduces to the classical second order method for $y'' = f(x,y)$.