Polynomial interpolation methods for viscous flow calculations

0. Summary.
Three subjects are considered in this paper. First, the notion of the resolving power of approximation methods, i.e. the number of intervals (or function values) per wavelength necessary to attain a preassigned error when approximating a given frequency, permits the evaluation of the relative efficiency of three types of spatial differencing used in setting up differential-difference equations for the 1-periodic parabolic problem ut = uxx. The three types of high-order spatial schemes considered are (explicit) centered differencing, the super-convergent smooth spline-Galerkin schemes discovered by Thomee, and the very high-order implicit schemes (Mehrstellenverfahren) which generalize Numerov's method. Seven time discretizations are introduced, namely Euler, backward-Euler, duFort-Frankel, 2nd order explicit, trapezoidal, Calahan-Zlamal, and 4th order Pade. The computational work necessary to solve each full discretization is minimized, for given error requirements, by balancing the number of intervals per wavelength against the number of time intervals per eth-life. The resulting data is used to compare the relative efficiency of these finite element and finite difference methods.

Secondly, some corresponding results for the hyperbolic problem ut = ux are briefly reviewed.

Finally, as Numerov-trapezoidal differencing turns out to be almost optimal for the heat equation, a tridiagonal implicit difference formula which extends Numerov differencing to the general 2nd order linear differential operator in one space dimension is presented. The technique used in deriving this scheme inspires certain difference analogs of some finite element schemes. It also leads to a curious modification of the diamond difference scheme for ut = ux. This “alternating kite” scheme has O(Δ2 + Δx3) truncation error with no more computational work than the diamond scheme itself.

The Construction and Comparison of Finite Difference Analogs of Some Finite Element Schemes

This thesis consists of two distinct parts which deal with two-point boundary value problems and parabolic problems, respectively. In Section 1 we examine the numerical solution of a two-point boundary value problem by a collocation method based on the consistency relationship of regular splines. An existence and convergence result is established which generalises the 0(h2) convergence result of the cubic spline collocation scheme for the problem in question. Contrary to most previously documented finite element schemes this method employs splines that may be non-linear in structure. Consequently, by a judicious choice of regular spline, the dominant terms of the true solution may be imitated more accurately than by the conventional polynomial based splines. The scheme is implemented by an algorithm that examines the suitability of various classes of regular splines and determines the subsequent deployment of them. The second section investigates semi-discrete finite element schemes for approximating the linear parabolic equation. A standard finite element discretization is employed for the space variable whilst an A0-stable, linear multistep, multiderivative discretization scheme, (L. M. S. D. ) is used in time. We consider both the homogeneous and the nonhomogeneous linear parabolic equations and derive optimal convergence results for the above schemes. The convergence results achieved with a k-step L. M. S. D. scheme, incorporating the first m derivatives, generalise and extend the studies of several authors who concentrate on the particular cases of linear multistep formulae, m-l, and one-step schemes, k=1. Ao-stable L. M. S. D. 's are constructed and their implementation procedures examined. The suitability of selecting a L. M. S. D. method, with m, k>1, in a semi-discrete Galerkin scheme is discussed, and its superiority over semi-discrete Galerkin schemes, that incorporate linear multistep or one-step formulae, is confirmed in several aspects. Finally, a class of quasi-linear parabolic equations is solved by a semi-discrete Galerkin scheme that is third order accurate in time. This method is based on a particular third order L. M. S. D. scheme and requires the solution of linearly algebraic systems of equations at each time level. Thus, we improve on all the previously documented linearised schemes as they are only second order accurate in time. All the schemes described in Section 2 are unconditionally stable.

/pdf/finite-element-solutions-to-boundary-value-problems-m3ineoh1kf.pdf

Finite element solutions to boundary value problems

The method of lines applied to a simple hyperbolic equation

Comparison of some fourth order difference schemes for hyperbolic problems

A new implicit finite-difference scheme for viscous flows is presented. The scheme is based on Simpson's rule and two-point Hermite interpolation, is uniformly accurate to fourth order in time and space, and is unconditionally stable according to a Fourier stability analysis. Numerical solutions of Burger's equation are presented to illustrate the order and accuracy of the scheme. 1. Introduction. The invention of the electronic digital computer stimulated the intensive development of numerical methods for the solution of fluid flow problems. The majority of the methods first developed were explicit and of low order because their utility on the small slow early computers derived from their simplicity. This trend has continued to the present, despite the accuracy of high order schemes and the unconditional stability of implicit schemes. The new parallel and pipeline com- puters, however, have spurred interest in complex difference methods by making feasible the large calculations required by such schemes. Recently Rusanov (5) and Burstein and Mirin (2) have studied explicit third order schemes for hyperbolic systems, while Zwas and Abarbanel (6) have developed an explicit fourth order method for special hyperbolic systems. We present in this paper a new unconditionally stable implicit scheme for viscous flows which is uniformly accurate to fourth order in time and space. We first describe the method and examine the local truncation error. We then present a linearized stability analysis of the scheme. Finally, we present several examples to illustrate the accuracy and stability of the method.

An implicit fourth order difference method for viscous flows

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An implicit fourth order difference method for viscous flows