On quasi-consistent integration by Nordsieck methods

In this paper we study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one important property for practical use, which has not attracted attention yet, i.e. the global error of a quasi-consistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion.     

Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation

Recently, Kulikov presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in such methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods.Here, we prove that the class of doubly quasi-consistent formulas is not empty and present the first example of such sort. This scheme belongs to the family of superconvergent explicit two-step peer methods constructed by Weiner, Schmitt, Podhaisky and Jebens. We present a sample of s-stage doubly quasi-consistent parallel explicit peer methods of order s−1 when s=3. The notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy conditions in automatic mode.     

Step size control in the numerical solution of stochastic differential equations

We introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations. The algorithm is based on a new pair of embedded explicit Runge-Kutta methods of strong order 1.5(1.0), where the method of strong order 1.5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error. We show that convergence of our method is preserved though the discretization times are not stopping times any more, and further, we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation.     

The numerical solution of one-phase classical Stefan problem

In this paper, variable space grid and boundary Immobilisation Techniques based on the explicit finite difference are applied to the one-phase classical Stefan problem. It is shown that all the results obtained by the two methods are in good agreement with the exact solution, and exhibit the expected convergence as the mesh size is refined.     

Nordsieck Methods with an Off-Step Point

Hybrid methods, incorporating one or more off-step points, are difficult to implement in a variable stepsize situation using the standard representation of input and output data in each step. However, instead of representing this data in terms of solution values and derivative values at a sequence of step points, it is possible to reformulate the method so that it operates on a Nordsieck vector. This has the consequence of reducing stepsize adjustments to nothing more than rescaling the components of the Nordsieck vector. This paper shows how to derive methods in both formulations and considers some implementation details. It is also possible to derive a new type of hybrid method using the Norsieck representation as the starting point and this is also discussed in the paper. The new method is found to have comparable accuracy for corresponding work expended as for standard methods.     

Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation

The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH ¹-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451.     

Probabilistic analysis of a differential equation for linear programming

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find the surprising result that the distribution of the convergence rate is a scaling function of a single variable. This scaling variable combines the convergence rate with the problem size (i.e., the number of variables and the number of constraints). We also estimate numerically the distribution of the computation time to an approximate solution, which is the time required to reach a vicinity of the attracting fixed point. We find that it is also a scaling function. Using the problem size dependence of the distribution functions, we derive high probability bounds on the convergence rates and on the computation times to the approximate solution.     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

Error analysis of staggered finite difference finite volume schemes on unstructured meshes

This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017     

Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.     

Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique

In this work we study the convergence of the fixed pivot techniques (Kumar and Ramkrishna Chem. Eng. Sci. 51, 1311–1332, 1996) for breakage problems. In particular, the convergence is investigated on four different types of uniform and non-uniform meshes. It is shown that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Furthermore, it gives first order convergence on a locally uniform mesh. Finally the analysis shows that the method does not converge on a non-uniform random mesh. The mathematical results of convergence analysis are also validated numerically.     

Invariant curves for variable step size integrators

The behaviour of one-step methods with variable step size applied to is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.The first author was partially supported by NSF Grant DMS87-19952 during his stay at UCLA.     

Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels

In this paper we investigate the attainable order of (global)convergence of collocation approximations in certain polynomialspline spaces for solution of Volterra integrodifferential equationswith weakly singular kernels. While the use of quasi-uniformmeshes leads, due to the nonsmooth nature of these solutions,to convergence of order less than one, regardless of the degreeof the approximating spline function, collocation on suitablygraded meshes will be shown to yield optimal convergence rates.     

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flow—so-called‘spurious solutions’ or ‘ghost solutions’.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems.     

Computing the Real Zeros of Hypergeometric Functions

Efficient methods for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. These methods are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.     

Error estimates for the approximation of semicoercive variational inequalities

Summary. An abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution. This condition turns out to be necessary as is demonstrated analytically and numerically. The results are applied to the finite element approximation of Poisson's equation with Signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions. For second order variational inequalities the condition is always satisfied, whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set. Applying the error estimate yields optimal order of convergence in terms of the mesh size . The numerical convergence rates observed are in good agreement with the predicted ones. Received August 16, 1993 / Revised version received March 21, 1994     

Scale and modify for the second and third order BDF methods

This paper investigates a ‘scale and modify’ technique used with variable stepsize BDF methods. When the stepsize is changed using the usual scaling procedure for Nordsieck methods, there can be adverse affects on the stability unless a severe restriction is placed on the allowable stepsize ratios. However, a modification to this scaling procedure may extend the range of permissible stepsize ratios. Results for the Nordsieck form of the second and third order methods indicate that this may be possible.     

A mixed collocation–finite difference method for 3D microscopic heat transport problems

Three-dimensional time-dependent initial-boundary value problems of a novel microscopic heat equation are solved by the mixed collocation–finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width. The collocation method on fixed grid size is used to approximate the space operator, whereas the finite difference scheme is used for time discretization. This new mixed method is applied to a novel heat problem in a particle, in order to compute the temperature distribution in and on the particle's surface. The second derivatives of the basis functions for the spectral approximation are derived. Direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method. The high-order accuracy and resolution achieved by the proposed method allows one to obtain engineering-accuracy solution on coarse meshes. The consistency, stability and convergence analysis are provided and numerical results are presented.     

Numerical solution to a one‐dimensional thermoplastic problem with unilateral constraint

In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ² + h²), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Linearly-implicit two-step methods and their implementation in Nordsieck form

We consider the Nordsieck representation of well-known linearly-implicit two-step methods in order to facilitate stepsize changes. These methods are treated as implicit schemes with one Newton step with a special predictor. A new predictor with better accuracy is discussed. Methods with 4–7 stages are tested and compared with RODAS. The numerical results show the potential of the Nordsieck implementation of the methods.