High‐order predictor–corrector algorithms

Abstract

SUMMARYNew predictor-corrector algorithms are presented for the computation of solution paths of non-linear partial di erential equations. The predictors and the correctors are based on perturbation techniques and Padà e approximants. This extends the Asymptotic Numerical Method (ANM), which is an e cient highorder continuation technique without corrector. The e ciency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier-Stokes equations. Many variants have been tested to establish an optimal algorithm.

Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants

“…Hence the path following procedure is efficient because the step number is very low for such a non-linear problem. Moreover, this step number can be reduced by choosing a larger , but in this case, the procedure should be completed by a correction phase [16]. Note also that the next example indicates that a strategy of large steps (i.e.…”

“…The first two examples present a linear pre-buckling and the last two tests present a non-linear one. These four tests will provide new validations of the path following technique based on Padé approximants, whose efficiency has been previously established [16,57]. The procedure presented in Section 5 extends the same Padé representation to initial post-bifurcation curves and this should lead to a significant reduction of the step number by comparison with Reference [40].…”

“…The perturbation ( U, ) is searched by the following asymptotic expansion: By substituting (16) and (17) into (13) and (14), one obtains the following sequence of linear problems:…”

“…Important improvements were realized: the use of Padé approximants permits to increase the validity range of the approximation and the continuation procedure permits to compute any equilibrium branch [13][14][15]. Recently, the high order predictor of the ANM has been associated with a high order corrector leading to a robust algorithm [16]. With respect to iterative algorithms, the advantages are the strong reduction of the number of matrix triangulations, the ability to compute a continuous branch with a very high accuracy and the reliability of the path following technique.…”

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

“…In the present paper, we consider only a speciÿc VPA [10], where all the fractions have a common denominator and with the same degrees for the numerators and the denominator. The interest of this procedure has been established by a number of practical applications for iterative algorithms and also for path following techniques; see [14] and references therein. These VPA's are deÿned as follows.…”