New multiple stages scheme with improved properties for second order problems

Abstract

A new multiple stages two–step scheme with best possible properties on phase and stability is introduced, for the first time in the literature. For this scheme we present a full theoretical, numerical and computational analysis. The ability of the new multiple stages scheme is examined by applying it on the solution of systems of coupled differential equations.

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Acknowledgements

The reported research was funded by Russian Foundation for Basic Research and the government of the Ulyanovsk region of the Russian Federation, Grant No. \(\text {18-48-730013}\).

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Correspondence to T. E. Simos.

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Supplement Material A

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$$\begin{aligned} LTE_{CL}= & {} LTE_{NM2S3SPD1} = LTE_{NM2S3SPD2} \\= & {} LTE_{NM2S3SPD3} = LTE_{NM2S3SPD4} = LTE_{NM2S3SPD5} \\\approx & {} h^{12} \, \vartheta _{0} = h^{12} \, \Biggl [ -{\frac{5\, ( iii (x) ) ^{3} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) }{598752}}-{\frac{23\,iii (x) \kappa (x) {\frac{{\mathrm{d}}^{8}}{{\mathrm{d}}{x}^{8}}}iii (x) }{11975040}}\\&\quad -\, {\frac{13\, ( iii (x) ) ^{3}\kappa (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) ^{2}}{1197504}}-{\frac{5\,iii (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) ^{3}}{199584}}\\&\quad -\, {\frac{ ( iii (x) ) ^{4} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) }{798336}}-{\frac{1201\, ( iii (x) ) ^{2}\kappa (x) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) \right) ^{2}}{23950080}}\\&\quad -\, {\frac{ \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}iii (x) \right) ^{2}\kappa (x) }{114048}}-{\frac{ \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) ^{4}\kappa (x) }{85536}}-{\frac{ ( iii (x) ) ^{6}\kappa (x) }{23950080}}\\&\quad -\, {\frac{157\, ( iii (x) ) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}iii (x) }{11975040}}-{\frac{743\,iii (x) \kappa (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) ^{2}{\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) }{5987520}}\\&\quad -\, {\frac{5\, ( iii (x) ) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) }{99792}}\\&\quad -\,{\frac{313\, ( iii (x) ) ^{2}\kappa (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) }{3991680}}\\&\quad -\, {\frac{23\,iii (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}iii (x) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) }{299376}}-{\frac{239\, ( iii (x) ) ^{2}\kappa (x) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}iii (x) }{23950080}}\\&\quad -\, {\frac{ \left( {\frac{{\mathrm{d}}^{10}}{{\mathrm{d}}{x}^{10}}}iii (x) \right) \kappa (x) }{23950080}}-{\frac{ \left( {\frac{{\mathrm{d}}^{9}}{{\mathrm{d}}{x}^{9}}}iii (x) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) }{2395008}}-{\frac{5\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) \right) ^{3}\kappa (x) }{177408}}\\ \end{aligned}$$
$$\begin{aligned}&\quad -\, {\frac{43\,iii (x) \kappa (x) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) \right) ^{2}}{748440}}-{\frac{iii (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}iii (x) }{187110}}\\&\quad -\,{\frac{ \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}iii (x) }{16632}}- {\frac{19\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}iii (x) }{443520}}\\&\quad -{\frac{37\, ( iii (x) ) ^{3}\kappa (x) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}iii (x) }{2993760}} -\, {\frac{19\, ( iii (x) ) ^{4}\kappa (x) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) }{4790016}}\\&\quad -{\frac{17\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) \right) \kappa (x) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}iii (x) }{1596672}}- {\frac{ \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) ^{2}\kappa (x) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}iii (x) }{19008}}\\&\quad -{\frac{31\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) \right) ^{2}}{266112}} - {\frac{7\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}iii (x) }{342144}}\\&\quad -{\frac{109\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) }{1197504}} - {\frac{31\, \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) \right) \kappa (x) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}iii (x) }{1995840}}\\&\quad -{\frac{13\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) \kappa (x) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}iii (x) }{2395008}}- {\frac{73\,iii (x) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\kappa (x) \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) }{598752}}\\&\quad -\,{\frac{323\,iii (x) \kappa (x) \left( {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}iii (x) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) }{5987520}}\\&\quad -\, {\frac{13\,iii (x) \kappa (x) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}iii (x) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) }{136080}}\\&\quad -\,{\frac{353\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}iii (x) \right) \kappa (x) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}iii (x) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}iii (x) }{2395008}} \Biggr ] \end{aligned}$$

where \(\kappa \left( x \right) = \kappa _{n}\).

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Kovalnogov, V.N., Fedorov, R.V., Suranov, D.V. et al. New multiple stages scheme with improved properties for second order problems. J Math Chem 57, 232–262 (2019). https://doi.org/10.1007/s10910-018-0948-8

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Keywords

  • Phase-lag
  • Derivative of the phase-lag
  • Initial value problems
  • Oscillating solution
  • Symmetric
  • Hybrid
  • Multistep
  • Schrödinger equation

Mathematical Subject Classification

  • 65L05