PPubMed20 Aug 2014Cited 11×

Implementing a seventh-order linear multistep method in a predictor-corrector mode or block mode: which is more efficient for the general second order initial value problem

Samuel N Jator, Leong Lee

Abstract

A Seventh-Order Linear Multistep Method (SOLMM) is developed and implemented in both predictor-corrector mode and block mode. The two approaches are compared by measuring their total number of function evaluations and CPU times. The stability property of the method is examined. This SOLMM is also compared with existing methods in the literature using standard numerical examples. AMS Subject Classification 65L05; 65L06

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The block mode implementation of the seventh-order linear multistep method is more accurate and uses fewer function evaluations than the predictor‑corrector mode.

What’s new

i.Developed a seventh‑order linear multistep method (SOLMM) for general second‑order IVPs.
ii.Implemented the SOLMM in both predictor‑corrector and block modes using Perl.
iii.Compared the two implementations, showing block mode superior in accuracy and function‑evaluation count.
iv.Benchmarked SOLMM against existing methods (VAR (8) and H8), demonstrating higher accuracy and efficiency.
v.Analyzed stability, finding a stability interval up to q = ‑4.552.

How they did it

The continuous form of the SOLMM was derived and its coefficients obtained by solving a 9 × 9 system with Mathematica. The resulting discrete method was coded in Perl and run on an AMD Quad‑Core A10‑4600M laptop, measuring NFEs and CPU times for several test problems.

What they found

For N=48, block mode achieved max error 1.92×10‑10 with 49 NFEs and CPU time 0.131 s, while predictor‑corrector had error 2.47×10‑5 with 91 NFEs.
In Table 4, SOLMM with 60 steps gave error 2.49×10‑8 versus VAR (8) error 7.1122×10‑7 with 67 steps.
Table 5 shows SOLMM required 362 NFEs to reach 1.7 correct decimal digits, compared with H8 needing 400 NFEs for only 0.3 digits.

Caveats

✕Only fixed‑step versions of the SOLMM were implemented; variable‑step versions are not yet available.
✕Performance was evaluated on a single laptop configuration, limiting generality of timing results.
✕Stability is guaranteed only for q in the interval [‑4.552, 0].

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