New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry

Medvedev, Maxim A.; Simos, T. E.

doi:10.1007/s10910-020-01168-5

New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry

Original Paper
Published: 29 September 2020

Volume 58, pages 2324–2360, (2020)
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New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry

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Abstract

A new FD (Finite Difference) method with zero phase-lag and its derivatives up to order six is proposed for the approximate solution of problems in Chemistry.

1 Introduction

1.1 The problem

We discuss the solution of the following Systems of Differential Equations:

$$\begin{aligned} \left[ \frac{d^{2}}{dx^{2}}+k_{i}^{2}-\frac{l_{i} (l_{i}+1)}{x^{2}}-POTENT_{ii} \right] \delta _{ij} = \sum _{m=1}^{N} POTENT_{im}\,\delta _{mj} \end{aligned}$$

(1)

where $1 \le i \le N$ and $m \ne i$.

(3) $\longrightarrow$ BVP ($=$ Boundary value problem) $\longrightarrow$ boundary conditions:

$$\begin{aligned} \delta _{ij}= & {} 0 \hbox { } at \hbox { } x = 0 \end{aligned}$$

(2)

$$\begin{aligned}&\delta _{ij} \sim k_{i}\, x j_{l_{i}}\,(k_{i} x) \theta _{ij} + \left( \frac{k_{i}}{k_{j}} \right) ^{1/2} K_{ij}\, k_{i}\, x\, n_{l{i}}\,(k_{i} x) \end{aligned}$$

(3)

where $j_{l}(x)$ and $n_{l}(x)$ $\longrightarrow$ spherical Bessel and Neumann functions respectively (see [62]).

Categories of problems (see [62]):

Open Channels
Close Channels

Asymptotic form (4) of (3) for the open channels problem (see [62]) ${\longrightarrow }$:

$$\begin{aligned} \varvec{\delta } \sim \varvec{\Xi } + \mathbf{T} \mathbf{W'}\,. \end{aligned}$$

where:

$$\begin{aligned} W'_{ij}= & {} \left( \frac{k_{i}}{k_{j}} \right) ^{1/2} K_{ij} \\ \Xi _{ij}= & {} k_{i} x j_{l_{i}}(k_{i} x) \theta _{ij} \\ T_{ij}= & {} k_{i} x n_{l_{i}}(k_{i} x) \theta _{ij} \end{aligned}$$

More details see in [67, 143,144,145,146,147,148,149,150,151] and references therein. Areas of application of the problems of the form (1) can also be found in [152,153,154,155,156,157,158,159,160] and references therein (Neural Networks, impulsive delayed systems, delayed switched systems, Impulsive control method, impulsive functional differential equations with infinite delays, singularly perturbed nonlinear systems, etc)

[62] $\longrightarrow$:

$$\begin{aligned} \left[ \frac{d^{2}}{dx^{2}} + k_{j' j}^{2}-\frac{l' (l' +1)}{x^{2}} \right] \delta _{j' l'}^{Jjl}(x)=\frac{2 \nu }{\hbar ^{2}} \sum _{j''} \sum _{l''} < j' l'; J \mid POTENT \mid j'' l'' ; J > \delta _{j'' l''}^{Jjl}(x) \end{aligned}$$

where (j, l), $(j ^\prime ,l ^\prime )$, $J=j+l=j ^\prime + l ^\prime$ $\longrightarrow$ [62].

and

$$\begin{aligned} k_{j' j}=\frac{2 \nu }{\hbar ^{2}} \left[ E + \frac{\hbar ^{2}}{2 I} \{ j(j+1)-j' (j' +1) \} \right] . \end{aligned}$$

where for E, I and $\nu$ $\longrightarrow$ [62] and references therein.

2 Introductiuon of the algorithm

In this section we will present the development of the new proposed scheme.

$$\begin{aligned} \overline{\{\beta \lambda \}}_{n-1}= & {} \{\beta \lambda \}_{n-1} - a_{0} \, h^2\, \left( \{\eta \sigma \}_{n+1} \right.\nonumber \\&\left.-\, 2 \, \{\eta \sigma \}_{n} + \{\eta \sigma \}_{n-1} \right) - 2\, a_{1} \, h^2\, \{\eta \sigma \}_{n} \nonumber \\ \overline{\overline{\{\beta \lambda \}}}_{n-1}= & {} \{\beta \lambda \}_{n-1} - a_{2} \, h^2\, \left( \{\eta \sigma \}_{n+1} \right.\nonumber \\ &\left.-\, 2 \, \{\eta \sigma \}_{n} + \overline{\{\eta \sigma \}}_{n-1} \right) - 2\, a_{3} \, h^2\, \{\eta \sigma \}_{n} \nonumber \\ \overline{\{\beta \lambda \}}_{n}= & {} \{\beta \lambda \}_{n} - a_{4} \, h^2\, \left( \{\eta \sigma \}_{n+1} - 2 \, \{\eta \sigma \}_{n} + \overline{\overline{\{\eta \sigma \}}}_{n-1} \right) \nonumber \\&\{\beta \lambda \}_{n+1} + a_{5} \, \{\beta \lambda \}_{n} + \{\beta \lambda \}_{n-1} \nonumber \\= & {} h^2 \left( b_{1} \, \left( \{\eta \sigma \}_{n+1} + \{\eta \sigma \}_{n-1} \right) + b_{0} \, \overline{\{\eta \sigma \}}_{n} \right) \end{aligned}$$

(4)

where $\{\eta \sigma \}_{n+i} = \{\beta \lambda \}''\left( t_{n+i}, \{\beta \lambda \}_{n+i} \right) , i=-1\left( 1 \right) 1$, $\overline{\{\eta \sigma \}}_{n-1} = \{\beta \lambda \}''\left( t_{n-1}, \overline{\{\beta \lambda \}}_{n-1} \right)$, $\overline{\overline{\{\eta \sigma \}}}_{n-1} = \{\beta \lambda \}''\left( t_{n-1}, \overline{\overline{\{\beta \lambda \}}}_{n-1} \right)$, $\overline{\{\eta \sigma \}}_{n} = \{\beta \lambda \}''\left( t_{n}, \overline{\{\beta \lambda \}}_{n} \right)$ and

$b_{{1}} = \frac{1}{12}$,
$a_{{j}}, \, j=0 (1) 5$ and $b_{0}$, to be determined.

Figure 1$\longrightarrow$ The Production of the of the new proposed algorithm (more information on flowcharts $\longrightarrow$ [161]).

In Fig. 1:

The general forms of the stability polynomials $ST_{k}, \, k=0,\,1$ is presented in [40]

Theorem 1${\mathop {\longrightarrow }\limits ^{using \, ST_{j}, \, \, j=0,1}}$: $PhL + DPL_{j}$ (with $PhL = {\text{phase}}{-}{\text{lag}}$ and $DPL_{j} = {\text{phase-lag's derivative of order}} j$

Theorem 1

[12] The terms OPL and $CF_{PL}$ are defined by:

$$\begin{aligned} -CF_{PL}\, v^{OPL+2} + O\left( v^{OPL+4} \right) = \frac{2\, ST_{1} \left( v \right) \, \cos \left( v \right) +ST_{0} \left( v\right) }{2\, ST_{1} \left( v\right) }. \end{aligned}$$

(8)

where OPL $\longrightarrow$ phase–lag’s order and $CF_{PL}$ $\longrightarrow$ phase–lag’s constant . OPL and $CF_{PL}$ ${\mathop {\longrightarrow }\limits ^{(4)- (7)}}$ phase–lag.

(4) $+$ $ST_{j}, \, \, j=0,1$ with general form given by [40] $+$ (8) $+$ Stage 5 of Fig. 1$\longrightarrow$:

$$\begin{aligned} a_{{0}}= & {} -\frac{1}{3}\,{\frac{PLM_{1}(v)}{{v}^{2} \, PLM_{0}(v)}}, \, a_{{1}} =-\frac{1}{6}\,{\frac{PLM_{2}(v)}{{v}^{2} \, PLM_{0}(v)}}, \nonumber \\ a_{{2}}= & {} -{\frac{PLM_{0}(v)}{{v}^{3}\,PLM_{3}(v)}}, \, a_{{3}} = -\frac{1}{3}\,{\frac{PLM_{4}(v)}{{v}^{3} \, PLM_{3}(v)}}, \nonumber \\ a_{{4}}= & {} -\frac{3}{2}\,{\frac{PLM_{3}(v)}{{v}^{2} \, PLM_{5}(v)}}, \, a_{{5}} = -\frac{1}{12}\,{\frac{PLM_{6}(v)}{PLM_{7}(v)}}, \nonumber \\ b_{{0}}= & {} \frac{1}{6}\,{\frac{PLM_{5}(v)}{v \, PLM_{7}(v)}} \end{aligned}$$

(9)

where

$$\begin{aligned} PLM_{0}(v)= & {} 12700800+743670\,\sin (v) {v}^{5}+1232\, (\cos (v))^{2}{v}^{8}+887\,\sin (v) {v}^{9} (\cos (v))^{2}\\&-\, 1680\,\cos (v) {v}^{7}\sin (v) -529200\,\cos (v) {v}^{3}\sin (v) \\&+\, 371835\,\sin (v) {v}^{5} (\cos (v))^{2}+3628800\,\cos (v) v\sin (v) \\&-\, 453600\,\sin (v) (\cos (v))^{2}v+14\,\sin (v) {v}^{11} (\cos (v))^{2}+27450\,\sin (v) {v}^{7} (\cos (v))^{2}\\&-\, 71280\,\cos (v) {v}^{5}\sin (v) -12436200\,{v}^{2}\\&-\, 189720\,{v}^{6}-200\,\sin (v) \cos (v) {v}^{9}+1025325\,\sin (v) {v}^{3} (\cos (v))^{2}\\&+\, 3432240\,{v}^{4}-12700800\,\cos (v) \\&+\, (\cos (v))^{3}{v}^{12}+44\,\cos (v) {v}^{12}+9261000\,\cos (v) {v}^{2}\\&-\, 601335\,\cos (v) {v}^{6}+191835\,\cos (v) {v}^{4}+2370\,\cos (v) {v}^{10}\\&-\, 11518\,\cos (v) {v}^{8}+1843\, (\cos (v))^{3}{v}^{8}\\&+\, 82440\, (\cos (v))^{3}{v}^{6}+14040\, (\cos (v))^{2}{v}^{6}+1509165\, (\cos (v))^{3}{v}^{4}\\&-\, 143640\, (\cos (v))^{2}{v}^{4}+7068600\, (\cos (v))^{3}{v}^{2}-3893400\, (\cos (v))^{2}{v}^{2}\\&-\, 250950\,\sin (v) {v}^{7}-3175200\,\sin (v) v\\&+\, 864675\,\sin (v) {v}^{3}+35\, (\cos (v))^{3}{v}^{10}\\&-\, 8\, (\cos (v))^{2}{v}^{10}+6828\,\sin (v) {v}^{9}+196\,\sin (v) {v}^{11}\\&+\, 48\,{v}^{10}-1232\,{v}^{8}-12700800\, (\cos (v))^{2}\\&+\, 12700800\, (\cos (v))^{3} \end{aligned}$$

$$\begin{aligned} PLM_{1}(v)= & {} -3175200+13230\,\sin (v) {v}^{5}+192\, (\cos (v))^{2}{v}^{8}\\&+\, 303\,\sin (v) {v}^{9} (\cos (v))^{2}-5760\,\cos (v) {v}^{7}\sin (v)\\&+\, 1360800\,\cos (v) {v}^{3}\sin (v) -186165\,\sin (v) {v}^{5} (\cos (v))^{2}\\&+\, 6350400\,\cos (v) v\sin (v) -7541100\,\sin (v) (\cos (v))^{2}v+6\,\sin (v) {v}^{11} (\cos (v))^{2}\\&+\, 3270\,\sin (v) {v}^{7} (\cos (v))^{2}-30240\,\cos (v) {v}^{5}\sin (v)\\&-\, 3250800\,{v}^{2}-196560\,{v}^{6}-240\,\sin (v) \cos (v) {v}^{9}\\&-\, 1828575\,\sin (v) {v}^{3} (\cos (v))^{2}+2308320\,{v}^{4}+3175200\,\cos (v) + (\cos (v))^{3}{v}^{12}\\&+\, 44\,\cos (v) {v}^{12}+1266300\,\cos (v) {v}^{2}\\&-\, 649215\,\cos (v) {v}^{6}-2218545\,\cos (v) {v}^{4}+3058\,\cos (v) {v}^{10}\\&+\, 24642\,\cos (v) {v}^{8}+4143\, (\cos (v))^{3}{v}^{8}\\&+\, 109620\, (\cos (v))^{3}{v}^{6}-15120\, (\cos (v))^{2}{v}^{6}\\&+\, 725445\, (\cos (v))^{3}{v}^{4}-418320\, (\cos (v))^{2}{v}^{4}+1908900\, (\cos (v))^{3}{v}^{2}\\&+\, 75600\, (\cos (v))^{2}{v}^{2}-229350\,\sin (v) {v}^{7}+1190700\,\sin (v) v+864675\,\sin (v) {v}^{3}\\&+\, 87\, (\cos (v))^{3}{v}^{10}-16\, (\cos (v))^{2}{v}^{10}\\&+\, 972\,\sin (v) {v}^{9}+84\,\sin (v) {v}^{11}+96\,{v}^{10}\\&+\, 2208\,{v}^{8}+3175200\, (\cos (v))^{2}-3175200\, (\cos (v))^{3} \end{aligned}$$

$$\begin{aligned}PLM_{2}(v)= & {} -6350400-524850\,\cos (v) {v}^{7}\sin (v) +2907\,\sin (v) {v}^{9} (\cos (v))^{2}\\&-\, 96948\,\sin (v) \cos (v) {v}^{9}-8745975\,\cos (v) {v}^{3}\sin (v) -295785\,\sin (v) {v}^{5} (\cos (v))^{2}\\&+\, 1587600\,\sin (v) (\cos (v))^{2}v+39690000\,\cos (v) v\sin (v) -22\,\sin (v) {v}^{11} (\cos (v))^{2}\\&-\, 2622375\,\sin (v) {v}^{3} (\cos (v))^{2}+16770\,\sin (v) {v}^{7} (\cos (v))^{2}-1313550\,\cos (v) {v}^{5}\sin (v) \\&-\, 3728025\,\sin (v) {v}^{3} (\cos (v))^{3}+453\,\sin (v) {v}^{9} (\cos (v))^{3}-248535\,\sin (v) {v}^{5} (\cos (v))^{3}\\&-\, 2370\,\sin (v) {v}^{7} (\cos (v))^{3}-1672\,\sin (v) \cos (v) {v}^{11}-14288400\,\sin (v) (\cos (v))^{3}v\\&+\, 22\,\sin (v) {v}^{11} (\cos (v))^{3}+ (\cos (v))^{3}{v}^{12}+44\,\cos (v) {v}^{12}+982800\,\cos (v) {v}^{2}\\&-\, 3608955\,\cos (v) {v}^{6}+11098395\,\cos (v) {v}^{4}\\&+\, 4458\,\cos (v) {v}^{10}+85038\,\cos (v) {v}^{8}\\&+\, 5667\, (\cos (v))^{3}{v}^{8}+245700\, (\cos (v))^{3}{v}^{6}-2401245\, (\cos (v))^{2}{v}^{6}\\&+\, 2434005\, (\cos (v))^{3}{v}^{4}-16268175\, (\cos (v))^{2}{v}^{4}+11718000\, (\cos (v))^{3}{v}^{2}\\&-\, 13381200\, (\cos (v))^{2}{v}^{2}+94650\,\sin (v) {v}^{7}\\&-\, 26989200\,\sin (v) v+15096375\,\sin (v) {v}^{3}\\&+\, 607\, (\cos (v))^{3}{v}^{10}-6384\, (\cos (v))^{2}{v}^{10}\\&-\, 18732\,\sin (v) {v}^{9}-308\,\sin (v) {v}^{11}\\&-\, 1770930\,\sin (v) {v}^{5}+110250\, (\cos (v))^{2}{v}^{8}\\&-\, 129\, (\cos (v))^{4}{v}^{10}-168\, (\cos (v))^{2}{v}^{12}\\&-\, 6621\, (\cos (v))^{4}{v}^{8}-147420\, (\cos (v))^{4}{v}^{6}-1285515\, (\cos (v))^{4}{v}^{4}\\&-\, 2343600\, (\cos (v))^{4}{v}^{2}+ (\cos (v))^{4}{v}^{12}\\&+\, 6350400\, (\cos (v))^{4}-12700800\, (\cos (v))^{3}\\&+\, 28808\,{v}^{10}+325416\,{v}^{8}+4021290\,{v}^{4}+3024000\,{v}^{2}-2706480\,{v}^{6}\\&+\, 392\,{v}^{12}+12700800\,\cos (v) \end{aligned}$$

$$\begin{aligned} PLM_{3}(v)= & {} (\cos (v))^{3}{v}^{11}+22\,\sin (v) (\cos (v))^{2}{v}^{10}\\&-\, 101\, (\cos (v))^{3}{v}^{9}+44\,\cos (v) {v}^{11}\\&+\, 791\,\sin (v) (\cos (v))^{2}{v}^{8}+308\,\sin (v) {v}^{10}\\&-\, 4725\, (\cos (v))^{3}{v}^{7}+2306\,\cos (v) {v}^{9}\\&+\, 23730\,\sin (v) (\cos (v))^{2}{v}^{6}+8764\,\sin (v) {v}^{8}\\&-\, 90720\, (\cos (v))^{3}{v}^{5}+14490\,\cos (v) {v}^{7}\\&+\, 466515\,\sin (v) (\cos (v))^{2}{v}^{4}-561750\,\sin (v) {v}^{6}-826875\, (\cos (v))^{3}{v}^{3}\\&+\, 719145\,\cos (v) {v}^{5}+4729725\,\sin (v) (\cos (v))^{2}{v}^{2}\\&-\, 1954890\,\sin (v) {v}^{4}-793800\, (\cos (v))^{3}v\\&-\, 9492525\,\cos (v) {v}^{3}+5556600\,\sin (v) (\cos (v))^{2}\\&+\, 10352475\,\sin (v) {v}^{2}+793800\,\cos (v) v\\&-\, 5556600\,\sin (v) \end{aligned}$$

$$\begin{aligned} PLM_{4}(v)= & {} 38102400-1351890\,\cos (v) {v}^{7}\sin (v) +2811\,\sin (v) {v}^{9} (\cos (v))^{2}\\&-\, 80124\,\sin (v) \cos (v) {v}^{9}+7130025\,\cos (v) {v}^{3}\sin (v) +987255\,\sin (v) {v}^{5} (\cos (v))^{2}\\&-\, 7484400\,\sin (v) (\cos (v))^{2}v+25174800\,\cos (v) v\sin (v) -14\,\sin (v) {v}^{11} (\cos (v))^{2}\\&-\, 1034775\,\sin (v) {v}^{3} (\cos (v))^{2}+118530\,\sin (v) {v}^{7} (\cos (v))^{2}+3591810\,\cos (v) {v}^{5}\sin (v) \\&+\, 2622375\,\sin (v) {v}^{3} (\cos (v))^{3}+1389\,\sin (v) {v}^{9} (\cos (v))^{3}+587385\,\sin (v) {v}^{5} (\cos (v))^{3}\\&+\, 39870\,\sin (v) {v}^{7} (\cos (v))^{3}-1064\,\sin (v) \cos (v) {v}^{11}-3402000\,\sin (v) (\cos (v))^{3}v\\&+\, 14\,\sin (v) {v}^{11} (\cos (v))^{3}+ (\cos (v))^{3}{v}^{12}\\&+\, 44\,\cos (v) {v}^{12}+58968000\,\cos (v) {v}^{2}\\&-\, 4564755\,\cos (v) {v}^{6}+16105635\,\cos (v) {v}^{4}\\&+\, 2634\,\cos (v) {v}^{10}-47418\,\cos (v) {v}^{8}\\&+\, 21363\, (\cos (v))^{3}{v}^{8}+388260\, (\cos (v))^{3}{v}^{6}-2287845\, (\cos (v))^{2}{v}^{6}\\&+\, 7027965\, (\cos (v))^{3}{v}^{4}-21560175\, (\cos (v))^{2}{v}^{4}+39009600\, (\cos (v))^{3}{v}^{2}\\&-\, 52390800\, (\cos (v))^{2}{v}^{2}+172890\,\sin (v) {v}^{7}\\&-\, 14288400\,\sin (v) v-8717625\,\sin (v) {v}^{3}\\&+\, 591\, (\cos (v))^{3}{v}^{10}-12744\, (\cos (v))^{2}{v}^{10}\\&-\, 12636\,\sin (v) {v}^{9}-196\,\sin (v) {v}^{11}\\&+\, 5719950\,\sin (v) {v}^{5}-224190\, (\cos (v))^{2}{v}^{8}\\&+\, 39\, (\cos (v))^{4}{v}^{10}-168\, (\cos (v))^{2}{v}^{12}\\&-\, 3549\, (\cos (v))^{4}{v}^{8}-157140\, (\cos (v))^{4}{v}^{6}-2429595\, (\cos (v))^{4}{v}^{4}\\&-\, 17803800\, (\cos (v))^{4}{v}^{2}+ (\cos (v))^{4}{v}^{12}-38102400\, (\cos (v))^{4}+76204800\, (\cos (v))^{3}\\&+\, 33240\,{v}^{10}+622344\,{v}^{8}+856170\,{v}^{4}-27783000\,{v}^{2}\\&-\, 182520\,{v}^{6}-76204800\,\cos (v) + 392\,{v}^{12} \end{aligned}$$

$$\begin{aligned} PLM_{5}(v)= & {} 14\, (\cos (v))^{3}{v}^{9}+13936\, (\cos (v))^{2}{v}^{9}\\&-\, 1350\, (\cos (v))^{3}{v}^{7}+307710\, (\cos (v))^{2}{v}^{7}\\&-\, 162540\, (\cos (v))^{3}{v}^{5}+3172365\, (\cos (v))^{2}{v}^{5}-3335850\, (\cos (v))^{3}{v}^{3}\\&-\, 32304825\, (\cos (v))^{2}{v}^{3}-3969000\, (\cos (v))^{3}v+15082200\, (\cos (v))^{2}v\\&-\, 88\,\cos (v) {v}^{11}+21240\,\cos (v) {v}^{7}+568890\,\cos (v) {v}^{5}\\&+\, 23785650\,\cos (v) {v}^{3}-3724\,\cos (v) {v}^{9}\\&-\, 14288400\,\cos (v) \sin (v) +3969000\,\cos (v) v\\&-\, 9525600\,v- (\cos (v))^{4}{v}^{11}+15330\,\sin (v) (\cos (v))^{3}{v}^{6}\\&-\, 1554\,\sin (v) (\cos (v))^{2}{v}^{8}-868770\,\sin (v) (\cos (v))^{2}{v}^{4}-28284\,\cos (v) {v}^{8}\sin (v) \\&-\, 392\,{v}^{11}+14288400\,\sin (v) +45360\,{v}^{5}-365400\,{v}^{7}\\&-\, 152\,\cos (v) {v}^{10}\sin (v) +2\,\sin (v) (\cos (v))^{3}{v}^{10}\\&+\, 339\,\sin (v) (\cos (v))^{3}{v}^{8}-962430\,\cos (v) {v}^{6}\sin (v) +14288400\,\sin (v) (\cos (v))^{3}\\&-\, 14288400\,\sin (v) (\cos (v))^{2}-2\, (\cos (v))^{3}{v}^{11}\\&+\, 168\, (\cos (v))^{2}{v}^{11}-25789050\,\sin (v) {v}^{2}\\&-\, 1566180\,\sin (v) {v}^{4}-336\,\sin (v) {v}^{8}\\&+\, 1267980\,\sin (v) {v}^{6}-135\, (\cos (v))^{4}{v}^{9}\\&-\, 7875\, (\cos (v))^{4}{v}^{7}-448\,\sin (v) {v}^{10}\\&-\, 283500\, (\cos (v))^{4}{v}^{5}-4843125\, (\cos (v))^{4}{v}^{3}\\&-\, 5556600\, (\cos (v))^{4}v+419895\,\sin (v) (\cos (v))^{3}{v}^{4}+12422025\,\sin (v) (\cos (v))^{3}{v}^{2}\\&+\, 26285175\,\cos (v) {v}^{2}\sin (v) -17173170\,\cos (v) {v}^{4}\sin (v) -12918150\,\sin (v) (\cos (v))^{2}{v}^{2}\\&-\, 49980\,\sin (v) (\cos (v))^{2}{v}^{6}-32\,\sin (v) (\cos (v))^{2}{v}^{10}-28616\,{v}^{9}+16698150\,{v}^{3} \end{aligned}$$

$$\begin{aligned} PLM_{6}(v)= & {} - (\cos (v))^{4}{v}^{12}- (\cos (v))^{3}{v}^{12}\\&+\, 44898\,\cos (v) {v}^{8}+513675\,\cos (v) {v}^{6}\\&+\, 9205245\,\cos (v) {v}^{4}-20752200\,\cos (v) {v}^{2}\\&-\, 906\,\cos (v) {v}^{10}+19051200\,\cos (v) -19051200\, (\cos (v))^{3}\\&-\, 44\,\cos (v) {v}^{12}+1053\, (\cos (v))^{4}{v}^{8}\\&-\, 2523\, (\cos (v))^{3}{v}^{8}+61020\, (\cos (v))^{4}{v}^{6}\\&-\, 95580\, (\cos (v))^{3}{v}^{6}+1295595\, (\cos (v))^{4}{v}^{4}-2787075\, (\cos (v))^{2}{v}^{6}\\&-\, 1607445\, (\cos (v))^{3}{v}^{4}+17917200\, (\cos (v))^{4}{v}^{2}-27972945\, (\cos (v))^{2}{v}^{4}\\&-\, 18257400\, (\cos (v))^{3}{v}^{2}+30618000\, (\cos (v))^{2}{v}^{2}+463590\,\sin (v) {v}^{7}\\&+\, 15082200\,\sin (v) v-25453575\,\sin (v) {v}^{3}\\&-\, 63\, (\cos (v))^{4}{v}^{10}+168\, (\cos (v))^{2}{v}^{12}\\&-\, 39\, (\cos (v))^{3}{v}^{10}+8784\, (\cos (v))^{2}{v}^{10}\\&+\, 6348\,\sin (v) {v}^{9}-140\,\sin (v) {v}^{11}\\&-\, 6086430\,\sin (v) {v}^{5}-111498\, (\cos (v))^{2}{v}^{8}+210105\,\sin (v) {v}^{5} (\cos (v))^{2}\\&-\, 26989200\,\cos (v) v\sin (v) -10\,\sin (v) {v}^{11} (\cos (v))^{2}\\&-\, 760\,\cos (v) \sin (v) {v}^{11}+1083\,\sin (v) {v}^{9} (\cos (v))^{3}\\&+\, 621495\,\sin (v) {v}^{5} (\cos (v))^{3}+6600825\,\sin (v) {v}^{3} (\cos (v))^{3}-392\,{v}^{12}+3177360\,{v}^{6}\\&+\, 7030800\,\sin (v) (\cos (v))^{3}v-622350\,\cos (v) {v}^{7}\sin (v) +39741975\,\cos (v) {v}^{3}\sin (v) \\&-\, 363\,\sin (v) {v}^{9} (\cos (v))^{2}-65388\,\cos (v) \sin (v) {v}^{9}\\&+\, 32130\,\sin (v) {v}^{7} (\cos (v))^{3}+15110550\,{v}^{4}\\&-\, 19051200\, (\cos (v))^{2}-19656\,{v}^{10}+200280\,{v}^{8}\\&-\, 2970\,\sin (v) {v}^{7} (\cos (v))^{2}-2153970\,\cos (v) {v}^{5}\sin (v) \\&+\, 4876200\,\sin (v) (\cos (v))^{2}v+10\,\sin (v) {v}^{11} (\cos (v))^{3}\\&-\, 9525600\,{v}^{2}+19051200\, (\cos (v))^{4}+4172175\,\sin (v) {v}^{3} (\cos (v))^{2} \end{aligned}$$

$$\begin{aligned} PLM_{7}(v)= & {} 793800+2338\,\cos (v) {v}^{8}-6030\,\cos (v) {v}^{6}\\&-\, 36855\,\cos (v) {v}^{4}+902475\,\cos (v) {v}^{2}\\&+\, 44\,\cos (v) {v}^{10}-793800\,\cos (v) +793800\, (\cos (v))^{3}\\&-\, 333\, (\cos (v))^{3}{v}^{8}-2925\, (\cos (v))^{3}{v}^{6}\\&-\, 6480\, (\cos (v))^{2}{v}^{6}+22680\, (\cos (v))^{3}{v}^{4}\\&+\, 279720\, (\cos (v))^{2}{v}^{4}+722925\, (\cos (v))^{3}{v}^{2}-340200\, (\cos (v))^{2}{v}^{2}\\&+\, 28620\,\sin (v) {v}^{7}-496125\,\sin (v) v\\&+\, 746550\,\sin (v) {v}^{3}+ (\cos (v))^{3}{v}^{10}+420\,\sin (v) {v}^{9}\\&+\, 236250\,\sin (v) {v}^{5}+8\, (\cos (v))^{2}{v}^{8}\\&-\, 32550\,\sin (v) {v}^{5} (\cos (v))^{2}+831600\,\cos (v) v\sin (v) \\&-\, 5040\,{v}^{6}+360\,\cos (v) {v}^{7}\sin (v) +630000\,\cos (v) {v}^{3}\sin (v) \\&+\, 30\,\sin (v) {v}^{9} (\cos (v))^{2}-430920\,{v}^{4}\\&-\, 793800\, (\cos (v))^{2}-48\,{v}^{8}-1305\,\sin (v) {v}^{7} (\cos (v))^{2}\\&-\, 58800\,\cos (v) {v}^{5}\sin (v) -335475\,\sin (v) (\cos (v))^{2}v\\&-\, 332325\,\sin (v) {v}^{3} (\cos (v))^{2}-1285200\,{v}^{2} \end{aligned}$$

where $PLM_{j}, \, \, j=0 \left( 1 \right) 7$ are the Polynomials of the Stability Functions of the Finite Difference Method. $v \rightarrow 0 \, \bigvee \, PLM_{0} (v) \rightarrow 0 \bigvee \, PLM_{3} (v)\bigvee \, PLM_{5} (v) \rightarrow 0\bigvee \, PLM_{7} (v) \rightarrow 0$ $\Longrightarrow$ expressions in Taylor series are procured:

$$\begin{aligned} a_{{0}}= & {} {\frac{6253}{844920}}+{\frac{3593135\,{v}^{2}}{13087979784}}+{\frac{155059456088527\,{v}^{4}}{14475309305738339520}}\nonumber \\&+\, {\frac{139425749259805708667053\,{v}^{6}}{332302096459882575766067328000}}+{\frac{12067246301799966527661025273\,{v}^{8}}{735052423458434275128783358533703680}}\nonumber \\&+\, {\frac{1260798970420267150992698787831456433\,{v}^{10}}{1964103811100132159983479296611888988328960000}}\nonumber \\&+\, {\frac{26931431246550687596648993340213858136431772243\,{v}^{12}}{1073115886322751758827494051328914422575848767101747200000}}\nonumber \\&+\, {\frac{97863671907202032375649481164664464470467236177873\,{v}^{14}}{99736678213900135767537890123370901131506475432956906864640000}}\nonumber \\&+\, {\frac{306830976068054013982599580863755706218398825647326804187343\,{v}^{16}}{7997387567236150128692781800561665211659139702173100865966367752192000000}}\nonumber \\&+\, \cdots \nonumber \\ a_{{1}}= & {} {\frac{9985}{14082}}-{\frac{28335125\,{v}^{2}}{1090664982}} +{\frac{203351534695147\,{v}^{4}}{241255155095638992}}\nonumber \\&-\, {\frac{27365377075618775531531\,{v}^{6}}{5538368274331376262767788800}} +{\frac{185855534073392276501922089063\,{v}^{8}}{612543686215361895940652798778086400}}\nonumber \\&+\, {\frac{37343460375933113598915017735122133\,{v}^{10}}{5036163618205467076880716145158689713664000}}\nonumber \\&+\, {\frac{5717010091236424020246422687686736347446346429\,{v}^{12}}{17885264772045862647124900855481907042930812785029120000}}\nonumber \\&+\, {\frac{12088619343930757998993122077959436749388380738011\,{v}^{14}}{977810570724511134975861667876185305210847798362322616320000}}\nonumber \\&+\, {\frac{129046489352538208519866214761253483454606765098381992672671\, {v}^{16}}{266579585574538337623092726685388840388637990072436695532212258406400000}}\nonumber \\&+\, \cdots \nonumber \\ a_{{2}}= & {} {\frac{2347}{173838}}+{\frac{706225\,{v}^{2}}{3392001558}} -{\frac{334471512575\,{v}^{4}}{842032719046954512}}\nonumber \\&-\, {\frac{7841623911295416464221\,{v}^{6}}{34089178296152633825813241600}} -{\frac{388086864125437095759628783\,{v}^{8}}{30222572340897565950959783695630080}}\nonumber \\&-\, {\frac{28696155154708289802564386864574089\,{v}^{10}}{56966544738901226599503666863763355170048000}}\nonumber \\&-\, {\frac{2372654593188563133854814744533991444773\,{v}^{12}}{149936855437299327738855912009605381528644207704000}}\nonumber \\&-\, {\frac{1643852457078370958422039614125061421413837943\,{v}^{14}}{4172787386362575557678066341577575651930196268455098368000}}\nonumber \\&-\, {\frac{1693438691000777535139766424322495017635497116807939871\,{v}^{16}}{262848336048573088926499022293076802538891393845397527788375224320000}}\nonumber \\&+\, {\frac{1284885800635756729106743771603107104181991930827236184217049\, {v}^{18}}{39038918922619090192343985661346398972644111027275100633987960339897876480000}}\nonumber \\&+\, \cdots \nonumber \\ a_{{3}}= & {} -{\frac{2149}{4139}}+{\frac{120575\,{v}^{2}}{17131321}} - {\frac{9105133539\,{v}^{4}}{67503023813288}}\nonumber \\&-\, {\frac{22798808971401383\,{v}^{6}}{14014453980650063445120}} + {\frac{284523356025137766286747\,{v}^{8}}{3727454316165015965634338956800}}\nonumber \\&+\, {\frac{380135844902730132032323513117\,{v}^{10}}{140517617540240565852674897605513881600}}\nonumber \\&+\, {\frac{24326982349154601537014420718636655849\,{v}^{12}}{192319198893441497821203671007991510698918336000}}\nonumber \\&+\, {\frac{20443381161218553287797991636756091101472479\,{v}^{14}}{5043514064497630821677711195897959002591966481772544000}}\nonumber \\&+\, {\frac{36116648956084847324390538204882996927562464153653\,{v}^{16}}{341242040673521525812068402413800367973665572692930886901760000}}\nonumber \\&+\, {\frac{300378001915884159863531799872583445903386549623805754807 \,{v}^{18}}{156481156495988015842327984853881669763684908719236414277649352011776000}} + \cdots \nonumber \\ a_{{4}}= & {} {\frac{4139}{1687400}}+{\frac{371\,{v}^{2}}{21902452}} -{\frac{913246049\,{v}^{4}}{4398025503071200}}\nonumber \\&-\, {\frac{98800064047693\,{v}^{6}}{13015692594809032128000}} +{\frac{54680106678648213863\,{v}^{8}}{415094646036686379361678080000}}\nonumber \\&+\, {\frac{249723269616349922681329\,{v}^{10}}{13121190595883896061784921953280000}}\nonumber \\&+\, {\frac{37634958285389933821373122383709\,{v}^{12}}{40311567048828011051023955807328478310400000}}\nonumber \\&+\, {\frac{11837158039143071446902882008083\,{v}^{14}}{459882545180086743259826027091026644162200000}}\nonumber \\&+\, {\frac{14784751989266194391946190194110556000523901\,{v}^{16}}{27583989318919433090894410135787926782401793810063360000000}}\nonumber \\&+\, {\frac{28881315212763976201711723501011223761848516633\,{v}^{18}}{3995728423972848535361073390157804552333020165585586126848000000}} + \cdots \nonumber \\ a_{{5}}= & {} -2-{\frac{53\,{v}^{16}}{96992075980800}}-{\frac{3605821607 \,{v}^{18}}{153374028035998049280000}} + \cdots \nonumber \\\ b_{{0}}= & {} \frac{5}{6}+{\frac{53\,{v}^{14}}{24248018995200}} +{\frac{17399693419\,{v}^{16}}{175284603469712056320000}}\nonumber \\&+\, {\frac{382402324300573\,{v}^{18}}{108936296047404976070688768000}} + \cdots \end{aligned}$$

(10)

$a_{{j}}, \, j=0 (1)5$ and $b_{{0}}$, for $v = \phi \, h$ ${\mathop {\longrightarrow }\limits ^{\text{are plotted in}}}$ Fig. 2

LTE ${\mathop {\longrightarrow }\limits ^{for \, the\, Method \, (4), \, which \, is \, declared \, as \, NM142S4SPD6}}$:

$$\begin{aligned} LTE_{NM142S4SPD6}= & {} \frac{53}{96992075980800}\, h^{16} \, \left(3\, \{\beta \lambda \}_{n}^{(16)} + 20 \, \phi ^{2} \, \{\beta \lambda \}_{n}^{(14)} \right.\nonumber \\&+\, 56 \, \phi ^{4} \, \{\beta \lambda \}_{n}^{(12)} + 84 \, \phi ^{6} \, \{\beta \lambda \}_{n}^{(10)} + 70 \, \phi ^{8} \, \{\beta \lambda \}_{n}^{(8)} \nonumber \\&\left.+\, 28\, \phi ^{10} \, \{\beta \lambda \}_{n}^{(6)} - 4\, \phi ^{14} \, \{\beta \lambda \}_{n}^{(2)} - \phi ^{16} \, \{\beta \lambda \}_{n} \right) + O \left( h^{18} \right) . \end{aligned}$$

(11)

3 Theoretical study of the new proposed method

3.1 Theoretical study of the local truncation error of the new proposed method

Local Truncation Error of the Finite Difference Method ${\mathop {\longrightarrow }\limits ^{scalar \, pattern}}$:

$$\begin{aligned} \{\beta \lambda \}''(x)= & {} \left( POTENT(x)-POTENT_{z} + \imath \right) \, \{\beta \lambda \}(x) \end{aligned}$$

(12)

with

POTENT(x) $\longrightarrow$ the potential.
$POTENT_{z}$ $\longrightarrow$ POTENT(x) on $x_{z}$.
$\hbox {PTN}(x) = POTENT(x) - POTENT_{z}$,
$\imath =POTENT_{z}-E$ and
E $\longrightarrow$ the energy.

For the contemplation of the Local Truncation Error of the Finite Difference Method, we use:

3.1.1 Classical algorithm, related as method A

$$\begin{aligned} LTE_{CL} = \frac{53}{32330691993600}\, h^{16} \, \{\beta \lambda \}_{n}^{(16)} + O \left( h^{18} \right) . \end{aligned}$$

(13)

3.1.2 Procedure projected in [40], related as method B

$$\begin{aligned} LTE_{NM142S4SPD1}= & {} \frac{53}{32330691993600}\, h^{16} \, \left( \{\beta \lambda \}_{n}^{(16)} + 4 \, \phi ^{6} \, \{\beta \lambda \}_{n}^{(10)} \right.\nonumber \\&\left.+\, 3 \, \phi ^{8} \, \{\beta \lambda \}_{n}^{(8)} \right) + O \left( h^{18} \right) . \end{aligned}$$

(14)

3.1.3 Procedure projected in [41], related as method C

$$\begin{aligned} LTE_{NM142S4SPD2}= & {} - \frac{53}{32330691993600}\, h^{16} \, \left(\{\beta \lambda \}_{n}^{(16)} - 6 \, \phi ^{4} \, \{\beta \lambda \}_{n}^{(12)} \right.\nonumber \\&\left.-\, 8 \, \phi ^{6} \, \{\beta \lambda \}_{n}^{(10)} - 3 \, \phi ^{8} \, \{\beta \lambda \}_{n}^{(8)} \right) + O \left( h^{18} \right) + O \left( h^{18} \right) . \end{aligned}$$

(15)

3.1.4 Procedure projected in [42], related as Method D

$$\begin{aligned} LTE_{NM142S4SPD3}= & {} - \frac{53}{32330691993600}\, h^{16} \, \left(\{\beta \lambda \}_{n}^{(16)} - 10 \, \phi ^{4} \, \{\beta \lambda \}_{n}^{(12)} \right.\nonumber \\&\left.-\, 20 \, \phi ^{6} \, \{\beta \lambda \}_{n}^{(10)} - 15 \, \phi ^{8} \, \{\beta \lambda \}_{n}^{(8)} - 4 \, \phi ^{10} \, \{\beta \lambda \}_{n}^{(6)} \right) + O \left( h^{18} \right) . \end{aligned}$$

(16)

3.1.5 Procedure projected in [43], related as Method E

$$\begin{aligned} LTE_{NM142S4SPD4}= & {} \frac{53}{32330691993600}\, h^{16} \, \left(\{\beta \lambda \}_{n}^{(16)} + 5 \, \phi ^{2} \, \{\beta \lambda \}_{n}^{(14)} \right.\nonumber \\&\left.+\, 10 \, \phi ^{4} \, \{\beta \lambda \}_{n}^{(12)} + 10 \, \phi ^{6} \, \{\beta \lambda \}_{n}^{(10)} + 5 \, \phi ^{8} \, \{\beta \lambda \}_{n}^{(8)} + \phi ^{10} \, \{\beta \lambda \}_{n}^{(6)} \right) + O \left( h^{18} \right) . \end{aligned}$$

(17)

3.1.6 Procedure projected in in [44], Related as Method F

$$\begin{aligned} LTE_{NM142S4SPD5}= & {} \frac{53}{193984151961600}\, h^{16} \, \left(6\, \{\beta \lambda \}_{n}^{(16)} + 35 \, \phi ^{2} \, \{\beta \lambda \}_{n}^{(14)} \right.\nonumber \\&+\, 84 \, \phi ^{4} \, \{\beta \lambda \}_{n}^{(12)} + 105 \, \phi ^{6} \, \{\beta \lambda \}_{n}^{(10)} + 70 \, \phi ^{8} \, \{\beta \lambda \}_{n}^{(8)} \nonumber \\&\left.+\, 21\, \phi ^{10} \, \{\beta \lambda \}_{n}^{(6)} + \phi ^{14} \, \{\beta \lambda \}_{n}^{(2)} \right) + O \left( h^{18} \right) . \end{aligned}$$

(18)

3.1.7 Procedure projected in Section 2, Related as Method G

The LTE of Method G is given by (11).

Figure 3$\longrightarrow$ flowchart for Local Truncation Error of the Finite Difference Method.

Stage 4.1 $\longrightarrow$:

$\imath \approx 0 \, \equiv \, E \approx V$: $POTENT_{z} \approx E \Rightarrow \imath =POTENT_{z} - E \approx 0 \Rightarrow \imath ^{i} \approx 0, \, i=1,2, \ldots \Rightarrow h^{\rho } \, \sum \limits _{i=0}^{\zeta }{\ell _{i}\, \imath ^{i}} \approx h^{\rho } \, \ell _{0}$.

Physical Causality:

$$\begin{aligned} LTE_{CL}= & {} LTE_{NM142S4SPD1}=LTE_{NM142S4SPD2}=\nonumber \\ LTE_{NM142S4SPD3}= & {} LTE_{NM142S4SPD4}=\nonumber \\ LTE_{NM142S4SPD5}= & {} LTE_{NM142S4SPD6} \approx h^{16} \, \ell _{0} \end{aligned}$$

(20)

$\ell _{0}$ $\longrightarrow$ Supplement Material A.

Conclusion 1

$\imath \approx 0$ $\longrightarrow$ $LTE_{CL}=LTE_{NM142S4SPD1}=LTE_{NM142S4SPD2}=LTE_{NM142S4SPD3} =LTE_{NM142S4SPD4}=LTE_{NM142S4SPD5}$.

Stage 4.2 $\longrightarrow$:

$\left| \imath \right|>>0$ $\equiv$ $\left| V \right|>> \bigvee<< \left| E \right|$: $E<<POTENT_{z} \vee E>>POTENT_{z} \Rightarrow \imath =POTENT_{z} - E>>0 \vee \imath =POTENT_{z} - E<<0$.

Conclusion 2

$\left| \imath \right|>>0$ $\longrightarrow$ $\min {LTE}=\sum \limits _{i=0}^{\daleth }{\ell _{i}\, \imath ^{i}}$ and $\daleth \longrightarrow \min {}$.

Physical Causality: Asymptotic forms of LTEs:

3.1.8 Method A

$$\begin{aligned} LTE_{CL} = \frac{53}{32330691993600}\, h^{16} \,\left( \{\beta \lambda \} (x) \, \imath ^{8} + \cdots \right) + O \left( h^{18} \right) . \end{aligned}$$

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3.1.9 Method B

$$\begin{aligned} LTE_{NM142S4SPD1}= & {} \frac{53}{16165345996800}\, h^{16} \, \left[\nonumber \right.\\&\left( 6\, \left( {\frac{\mathrm{d}}{\mathrm{d}{x}}}{\text{PTN}} (x) \right) \, \left( \frac{{\mathrm{d}}}{{\mathrm{d}}{x}}\{\beta \lambda \} (x) \right) + 59\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x)\right. \nonumber \\&\left.\left.+\, 3\, {\text{PTN}} (x)^{2}\, \{\beta \lambda \} (x)\right) \imath ^{6} + \cdots \right] + O \left( h^{18} \right) . \end{aligned}$$

(22)

3.1.10 Method C

$$\begin{aligned} LTE_{NM142S4SPD2}= & {} \frac{53}{2020668249600}\, h^{16} \nonumber \\&\left[ \left(\left( {\frac{{\mathrm{d}}^{2}}{{{\mathrm{d}}}{x}^{2}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \right) \imath ^{6} + \cdots \right] + O \left( h^{18} \right) . \end{aligned}$$

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3.1.11 Method D

$$\begin{aligned} LTE_{NM142S4SPD3}= & {} \frac{53}{8082672998400}\, h^{16} \nonumber \\&\left[ \left(10\, \left( {\frac{{\mathrm{d}}{^{3}}}{{\mathrm{d}}{x}^{3}}}{\text{PTN}} (x) \right) \, \left( \frac{\mathrm{d}}{{\mathrm{d}}{x}}\{\beta \lambda \} (x) \right) \right.\right.\nonumber \\&+\, 20\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \, {\text{PTN}} (x) \nonumber \\&+\, 79\, \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \nonumber \\&+\left.\left.\, 15\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}{x}}}{\text{PTN}} (x) \right)^2\, \{\beta \lambda \} (x) \right) \imath ^{5} + \cdots \right] + O \left( h^{18} \right) . \end{aligned}$$

(24)

3.1.12 Method E

$$\begin{aligned} LTE_{NM142S4SPD4}= & {} \frac{53}{2020668249600}\, h^{16} \nonumber \\&\left[ \left(\left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \right) \imath ^{5} + \cdots \right] + O \left( h^{18} \right) . \end{aligned}$$

(25)

3.1.13 Method F

$$\begin{aligned} LTE_{NM142S4SPD5}= & {} \frac{1537}{351596275430400}\, h^{16} \nonumber \\ &\left[ \left(87\, \left( {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \right.\right.\nonumber \\&+\, 14\, \left( {\frac{{\mathrm{d}}{^{5}}}{{{\mathrm{d}}}{x}^{5}}}{\text{PTN}} (x) \right) \, \left( \frac{{{\mathrm{d}}}}{{{\mathrm{d}}}{x}}\{\beta \lambda \} (x) \right) \nonumber \\&+\, 70\, \left( {\frac{{{\mathrm{d}}}^{2}}{{{\mathrm{d}}}{x}^{2}}}{\text{PTN}} (x) \right)^2\, \{\beta \lambda \} (x) \nonumber \\&+\, 105\, \left( {\frac{{{\mathrm{d}}}^{3}}{{{\mathrm{d}}}{x}^{3}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \, \left( {\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}{x}}}{\text{PTN}} (x) \right) \nonumber \\&\left.\left.+\, 42\, \left( {\frac{{{\mathrm{d}}}^{4}}{{{\mathrm{d}}}{x}^{4}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \, {\text{PTN}} (x) \right) \imath ^{4} + \cdots \right] + O \left( h^{18} \right) . \end{aligned}$$

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3.1.14 Method G

$$\begin{aligned} LTE_{NM142S4SPD6}= & {} \frac{53}{378875296800}\, h^{16} \nonumber \\&\left[\left ( \left( {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}{\text{PTN}} (x) \right) \, \{\beta \lambda \} (x) \right) \, \imath^{4} + \cdots \right] + O \left( h^{18} \right) . \end{aligned}$$

(27)

The above theoretical results give:

Theorem 2

Method A: $LTE=O\left( \imath ^8 \right)$.
Method B: $LTE=ADSX\,\, \, O\left( \imath ^6 \right)$.
Method C: $LTE=BYSP\,\, \, O\left( \imath ^6 \right)$.
Method D: $LTE=TNSI\,\, \, O\left( \imath ^5 \right)$.
Method E: $LTE=VCSM\,\, \, O\left( \imath ^5 \right)$.
Method F: $LTE=STVY\,\, \, O\left( \imath ^4 \right)$.
Method F: $LTE=AZCC\,\, \, O\left( \imath ^4 \right)$.

with ${\text{BYSP}}<< {\text{ADSX}}$, ${\text{VCSM}}<< {\text{TNSI}}$ and ${\text{AZCC}}<< {\text{STVY}}$. Theorem 2$\longrightarrow$ Method G is the most effectual one on accuracy.

3.2 Theoretical study of the stability (stability analysis of the new introduced method

Stability Analysis of the Finite Difference Method ${\mathop {\longrightarrow }\limits ^{{\text{for}} \, {\text{the}} \, {\text{newly}} \, {\text{Produced}} \, {\text{Scheme}}}}$ scholar’s paragon:

$$\begin{aligned} \{\beta \lambda \}''=-\chi ^{2} \, \{\beta \lambda \}\,. \end{aligned}$$

(28)

Conclusion 3

(5), (28) $\longrightarrow$ $\chi \ne \phi$.

Figure 4$\longrightarrow$ Stability Analysis of the New Introduced Finite Difference Method.

Stage 1 $+$ Stage 2 $\longrightarrow$:

$$\begin{aligned} ST_{1} \left( s,v\right) \, \left( \{\beta \lambda \}_{n+1} + \{\beta \lambda \}_{n-1} \right) + ST_{0} \left( s,v\right) \, \{\beta \lambda \}_{n} = 0 \end{aligned}$$

(29)

and

$$\begin{aligned} ST_{1} \left( s,v\right) \, \left( \sigma ^2 + 1 \right) + ST_{0} \left( s,v\right) \, \sigma = 0 \end{aligned}$$

(30)

where $ST_{j} \left( s,v\right) , \, j=0,1$ are given in [40]

Stage 3 $\longrightarrow$: $s = \chi \, h$ $\wedge$ $v = \phi \, h$.

Stage 4 $\longrightarrow$:

$$\begin{aligned} ST_{1} \left( s,v\right) = {\frac{PLM_{9}\left( s,v \right) }{PLM_{8}\left( s,v \right) }}, \, ST_{0} \left( s,v\right) = \frac{1}{2} \, {\frac{PLM_{10}\left( s,v \right) }{PLM_{8}\left( s,v \right) }} \end{aligned}$$

(31)

where

$$\begin{aligned} PLM_{8}\left( s,v \right)= & {} -3987900\,{v}^{11}\sin (3\,v)-27839700\,\sin (v) {v}^{9} \\&-\, 13970880\,{v}^{12}+272160\,{v}^{12}\cos (3\,v) -1411200\,{v}^{13}\sin (2\,v) +15120000\,{v}^{11}\sin (2\,v) \\&-\, 952560\,\cos (v) {v}^{12}+69344100\,\cos (v) {v}^{10}-9525600\,\cos (v) {v}^{8}\\&+\, 31846500\,\sin (v) {v}^{11}-69854400\,{v}^{10}+19051200\,{v}^{8}+19958400\,{v}^{9}\sin (2\,v) \\&-\, 19051200\,{v}^{8}\cos (2\,v) +8640\,{v}^{15}\sin (2\,v) \\&-\, 155520\,{v}^{14}\cos (2\,v) +6713280\,{v}^{12}\cos (2\,v) \\&-\, 35100\,{v}^{14}\cos (3\,v) -8164800\,{v}^{10}\cos (2\,v) \\&+\, 360\,{v}^{17}\sin (3\,v) -390600\,{v}^{13}\sin (3\,v) \\&+\, 12\,{v}^{18}\cos (3\,v) +192\,{v}^{16}\cos (2\,v) \\&-\, 3996\,{v}^{16}\cos (3\,v) -4025700\,{v}^{9}\sin (3\,v) +100236\,\cos (v) {v}^{16}\\&-\, 394740\,\cos (v) {v}^{14}+1358100\,\sin (v) {v}^{15}+8675100\,{v}^{10}\cos (3\,v) -397440\,{v}^{14}\\&+\, 10949400\,\sin (v) {v}^{13}-15660\,{v}^{15}\sin (3\,v) \\&+\, 20520\,\sin (v) {v}^{17}+2148\,\cos (v) {v}^{18}\\&+\, 9525600\,{v}^{8}\cos (3\,v) -2112\,{v}^{16}\\ PLM_{9}\left( s,v \right)= & {} 537\,\cos (v) {s}^{6}{v}^{14}-3987900\,{v}^{11}\sin (3\,v) \\&-\, 13970880\,{v}^{12}+272160\,{v}^{12}\cos (3\,v) \\&-\, 2381400\,\cos (v) {s}^{4}{v}^{6}-12493\,\cos (v) {s}^{8}{v}^{10}\\&-\, 26763\,\cos (v) {s}^{4}{v}^{14}+6697845\,\cos (v) {s}^{8}{v}^{4}\\&-\, 793800\,\cos (v) {s}^{2}{v}^{8}-110997\,\cos (v) {s}^{8}{v}^{8}\\&-\, 1411200\,{v}^{13}\sin (2\,v) +15120000\,{v}^{11}\sin (2\,v) \\&-\, 131355\,\cos (v) {s}^{4}{v}^{12}-121629\,\cos (v) {s}^{6}{v}^{10}\\&-\, 952560\,\cos (v) {v}^{12}+69344100\,\cos (v) {v}^{10}\\&-\, 39463200\,\sin (v) {s}^{6}{v}^{3}-107541\,\sin (v) {s}^{4}{v}^{13}\\&+\, 113175\,\sin (v) {s}^{2}{v}^{15}-2929050\,\sin (v) {s}^{6}{v}^{9}\\&-\, 9525600\,\cos (v) {v}^{8}-27839700\,\sin (v) {v}^{9}\\&+\, 31846500\,\sin (v) {v}^{11}+2268000\,\cos (v) {s}^{8}{v}^{6}\\&+\, 174749400\,\cos (v) {s}^{6}{v}^{4}-10791900\,\cos (v) {s}^{8}{v}^{2}\\&-\, 69854400\,{v}^{10}+19051200\,{v}^{8}\\&+\, 19958400\,{v}^{9}\sin (2\,v) -19051200\,{v}^{8}\cos (2\,v) \\&+\, 84597\,\sin (v) {s}^{6}{v}^{11}-79380\,\cos (v) {s}^{2}{v}^{12}\\&+\, 8640\,{v}^{15}\sin (2\,v) +2778300\,\sin (v) {s}^{8}v\\&+\, 914130\,\sin (v) {s}^{8}{v}^{7}-3762\,\sin (v) {s}^{4}{v}^{15}\\&+\, 1710\,\sin (v) {s}^{2}{v}^{17}+3175200\,{s}^{8}\cos (3\,v) \\&-\, 6350400\,{s}^{8}\cos (2\,v) -155520\,{v}^{14}\cos (2\,v) \\&-\, 151200\,{s}^{8}{v}^{2}\cos (2\,v) -{v}^{12}{s}^{8}\cos (3\,v) \\&+\, 133245\,\sin (v) {s}^{8}{v}^{5}+60480\,{s}^{8}{v}^{5}\sin (2\,v) \\&+\, 6713280\,{v}^{12}\cos (2\,v) +3\,{v}^{14}{s}^{6}\cos (3\,v) \\&-\, 3\,{s}^{4}{v}^{16}\cos (3\,v) +{v}^{18}{s}^{2}\cos (3\,v) -87\,{s}^{8}{v}^{10}\cos (3\,v) \\&-\, 35100\,{v}^{14}\cos (3\,v) +21205800\,{s}^{6}{v}^{4}\cos (3\,v) \\&-\, 14189175\,{s}^{4}{v}^{7}\sin (3\,v) +1587600\,{s}^{2}{v}^{8}\\&+\, 50009400\,\sin (v) {s}^{4}{v}^{5}-8164800\,{v}^{10}\cos (2\,v) \\&+\, 21772800\,{s}^{6}{v}^{3}\sin (2\,v) -32895\,\cos (v) {s}^{2}{v}^{14}\\&-\, 2925\,{s}^{2}{v}^{14}\cos (3\,v) +186165\,{s}^{8}{v}^{5}\sin (3\,v) \\&+\, 14175\,{s}^{4}{v}^{12}\cos (3\,v) -384\,{s}^{8}{v}^{8}\cos (2\,v) \\&+\, 5778675\,\cos (v) {s}^{2}{v}^{10}+5529\,{s}^{6}{v}^{10}\cos (3\,v) \\&-\, 16669800\,{s}^{4}{v}^{5}\sin (3\,v) -4143\,{s}^{8}{v}^{8}\cos (3\,v) \\&+\, 360\,{v}^{17}\sin (3\,v) +38102400\,{s}^{6}{v}^{2}\cos (3\,v) \\&+\, 2661\,{s}^{6}{v}^{11}\sin (3\,v) +16\,{s}^{2}{v}^{16}\cos (2\,v) \\&-\, 5821200\,{s}^{2}{v}^{10}-1908900\,{s}^{8}{v}^{2}\cos (3\,v) \\&+\, 1260000\,{s}^{2}{v}^{11}\sin (2\,v) -390600\,{v}^{13}\sin (3\,v) \\&-\, 48\,{s}^{6}{v}^{12}\cos (2\,v) -12700800\,{s}^{8}v\sin (2\,v) \\&-\, 66\,{s}^{4}{v}^{15}\sin (3\,v) -76204800\,{s}^{6}{v}^{2}\cos (2\,v) \\&+\, 32\,{s}^{8}{v}^{10}\cos (2\,v) +2480625\,{s}^{4}{v}^{8}\cos (3\,v) \\&-\, 1360800\,{s}^{6}{v}^{3}\sin (3\,v) -333\,{s}^{2}{v}^{16}\cos (3\,v) \\&+\, 303\,{s}^{4}{v}^{14}\cos (3\,v) +105\,{s}^{6}{v}^{12}\cos (3\,v) \\&+\, 12\,{v}^{18}\cos (3\,v) +192\,{v}^{16}\cos (2\,v) -3996\,{v}^{16}\cos (3\,v) \\&-\, 7392\,{s}^{6}{v}^{10}+720\,{s}^{2}{v}^{15}\sin (2\,v) -4025700\,{v}^{9}\sin (3\,v) \\&+\, 7541100\,{s}^{8}v\sin (3\,v) +722925\,{s}^{2}{v}^{10}\cos (3\,v) \\&+\, 100236\,\cos (v) {v}^{16}-9216\,{s}^{8}{v}^{8}\\&-\, 10080\,{s}^{6}{v}^{9}\sin (2\,v) -1587600\,{s}^{2}{v}^{8}\cos (2\,v) \\&+\, 42\,{s}^{6}{v}^{13}\sin (3\,v) -3270\,{s}^{8}{v}^{7}\sin (3\,v) \\&-\, 680400\,{s}^{2}{v}^{10}\cos (2\,v) -725445\,{s}^{8}{v}^{4}\cos (3\,v) \\&-\, 3175200\,\cos (v) {s}^{8}+82350\,{s}^{6}{v}^{9}\sin (3\,v) \\&-\, 32550\,{s}^{2}{v}^{13}\sin (3\,v) -4191\,\sin (v) {s}^{8}{v}^{9}\\&+\, 8353\,\cos (v) {s}^{2}{v}^{16}-394740\,\cos (v) {v}^{14}\\&+\, 1358100\,\sin (v) {v}^{15}-303\,{s}^{8}{v}^{9}\sin (3\,v) \\&+\, 22680\,{s}^{2}{v}^{12}\cos (3\,v) -6\,{s}^{8}{v}^{11}\sin (3\,v) \\&-\, 1164240\,{s}^{2}{v}^{12}-33120\,{s}^{2}{v}^{14}+528\,{s}^{6}{v}^{12}\\&-\, 176\,{s}^{2}{v}^{16}-861840\,{s}^{6}{v}^{6}\cos (2\,v) \\&-\, 427680\,{s}^{6}{v}^{7}\sin (2\,v) -1200\,{s}^{6}{v}^{11}\sin (2\,v) \\&-\, 117600\,{s}^{2}{v}^{13}\sin (2\,v) +1663200\,{s}^{2}{v}^{9}\sin (2\,v) \\&-\, 71190\,{s}^{4}{v}^{11}\sin (3\,v) +8675100\,{v}^{10}\cos (3\,v) \\&-\, 2319975\,\sin (v) {s}^{2}{v}^{9}-3175200\,{s}^{6}{v}^{5}\sin (2\,v) \\&-\, 335475\,{s}^{2}{v}^{9}\sin (3\,v) -2721600\,{s}^{8}{v}^{3}\sin (2\,v) \\&+\, 3075975\,{s}^{6}{v}^{5}\sin (3\,v) +1828575\,{s}^{8}{v}^{3}\sin (3\,v) \\&-\, 12960\,{s}^{2}{v}^{14}\cos (2\,v) +6669810\,\sin (v) {s}^{4}{v}^{11}\\&+\, 7392\,{s}^{6}{v}^{10}\cos (2\,v) +28755\,\cos (v) {s}^{6}{v}^{12}\\&+\, 11520\,{s}^{8}{v}^{7}\sin (2\,v) -1399545\,{s}^{4}{v}^{9}\sin (3\,v) \\&-\, 2373\,{s}^{4}{v}^{13}\sin (3\,v) -138418875\,\sin (v) {s}^{4}{v}^{7}\\&+\, 2653875\,\sin (v) {s}^{2}{v}^{11}+1115505\,{s}^{6}{v}^{7}\sin (3\,v) \\&-\, 1305\,{s}^{2}{v}^{15}\sin (3\,v) +559440\,{s}^{2}{v}^{12}\cos (2\,v) \\&-\, 397440\,{v}^{14}+30240\,{s}^{8}{v}^{6}\cos (2\,v) \\&+\, 40325040\,{s}^{6}{v}^{6}-172594800\,{s}^{6}{v}^{4}\\&-\, 2192400\,{s}^{6}{v}^{8}+30\,{s}^{2}{v}^{17}\sin (3\,v) -332325\,{s}^{2}{v}^{11}\sin (3\,v) \\&+\, 10949400\,\sin (v) {v}^{13}-38102400\,\cos (v) {s}^{6}{v}^{2}\\&-\, 537\,\cos (v) {s}^{4}{v}^{16}+179\,\cos (v) {s}^{2}{v}^{18}\\&-\, 352\,{s}^{8}{v}^{10}+4527495\,{s}^{6}{v}^{6}\cos (3\,v) \\&+\, 793800\,{s}^{2}{v}^{8}\cos (3\,v) -15660\,{v}^{15}\sin (3\,v) \\&+\, 76204800\,{s}^{6}{v}^{2}+121352175\,\cos (v) {s}^{4}{v}^{8}\\&+\, 22059135\,\sin (v) {s}^{4}{v}^{9}+10039545\,\sin (v) {s}^{6}{v}^{7}\\&+\, 13452075\,\sin (v) {s}^{6}{v}^{5}-1630125\,\sin (v) {s}^{8}{v}^{3}\\&-\, 179\,\cos (v) {s}^{8}{v}^{12}+20520\,\sin (v) {v}^{17}\\&+\, 12852000\,{s}^{8}{v}^{2}+2148\,\cos (v) {v}^{18}\\&+\, 272160\,{s}^{4}{v}^{10}\cos (3\,v) -8396640\,{s}^{8}{v}^{4}\\&+\, 816480\,{s}^{8}{v}^{6}+2394\,\sin (v) {s}^{6}{v}^{13}-342\,\sin (v) {s}^{8}{v}^{11}\\&+\, 480\,{s}^{8}{v}^{9}\sin (2\,v) -109620\,{s}^{8}{v}^{6}\cos (3\,v) \\&+\, 15884505\,\cos (v) {s}^{6}{v}^{6}-6474060\,\cos (v) {s}^{6}{v}^{8}\\&+\, 9525600\,{v}^{8}\cos (3\,v) -7813260\,\cos (v) {s}^{4}{v}^{10}\\&+\, 836640\,{s}^{8}{v}^{4}\cos (2\,v) -2112\,{v}^{16}+6350400\,{s}^{8}\\&+\, 2381400\,{s}^{4}{v}^{6}\cos (3\,v) +84240\,{s}^{6}{v}^{8}\cos (2\,v) \\&+\, 247320\,{s}^{6}{v}^{8}\cos (3\,v) -23360400\,{s}^{6}{v}^{4}\cos (2\,v) \\&+\, 912450\,\sin (v) {s}^{2}{v}^{13} \end{aligned}$$

$$\begin{aligned} PLM_{10}\left( s,v \right)= & {} -35607600\,{s}^{6}{v}^{4}\cos (4\,v) +191160\,{v}^{14}\cos (3\,v) +668\,{v}^{12}{s}^{8}\cos (2\,v)\\&-\, 420210\,{v}^{13}\sin (3\,v) +36514800\,{v}^{10}\cos (3\,v) \\&+\, 78\,{s}^{6}{v}^{12}\cos (4\,v) -7098\,{s}^{6}{v}^{10}\cos (4\,v) \\&-\, 621495\,{v}^{13}\sin (4\,v) +7151760\,\cos (v) {s}^{2}{v}^{12}\\&+\, 5244750\,{s}^{6}{v}^{5}\sin (4\,v) -8344350\,{v}^{11}\sin (3\,v) \\&-\, 57153600\,{s}^{2}{v}^{7}\sin (2\,v) -9686250\,{s}^{2}{v}^{10}\cos (4\,v) \\&+\, 6644\,{v}^{11}{s}^{8}\sin (2\,v) +20087760\,\sin (v) {s}^{2}{v}^{13}\\&-\, 42480720\,\cos (v) {s}^{6}{v}^{8}+18325440\,\cos (v) {s}^{8}{v}^{6}\\&+\, 259386\,{v}^{17}\sin (2\,v) +147420\,{s}^{8}{v}^{6}\cos (4\,v) \\&+\, 2148\,\cos (v) {s}^{4}{v}^{16}-1432\,\cos (v) {s}^{6}{v}^{14}\\&-\, 8456\,{v}^{13}{s}^{6}\sin (2\,v) +106709400\,{v}^{12}\cos (2\,v) \\&-\, 6600825\,{v}^{11}\sin (4\,v) -6350400\,{s}^{8}\cos (4\,v) \\&-\, 1432\,\cos (v) {s}^{2}{v}^{18}+3876\,\sin (v) {s}^{8}{v}^{11}\\&+\, 31253040\,\cos (v) {s}^{4}{v}^{10}+6621\,{s}^{8}{v}^{8}\cos (4\,v) \\&-\, 453\,{s}^{8}{v}^{9}\sin (4\,v) -34884\,{v}^{18}\cos (2\,v) \\&+\, 7372890\,{v}^{13}\sin (2\,v) +55310040\,\sin (v) {s}^{6}{v}^{7}\\&+\, 1174770\,{s}^{6}{v}^{7}\sin (4\,v) +14956560\,\sin (v) {s}^{6}{v}^{9}\\&-\, 1053\,{v}^{16}\cos (4\,v) -26679240\,\sin (v) {s}^{4}{v}^{11}\\&+\, 6350400\,{s}^{8}-4859190\,{s}^{6}{v}^{6}\cos (4\,v) \\&-\, 4447260\,\sin (v) {s}^{8}{v}^{7}-1208\,{v}^{17}{s}^{2}\sin (2\,v) \\&+\, 12700800\,{s}^{8}\cos (3\,v) -1083\,{v}^{17}\sin (4\,v) \\&+\, 839790\,{s}^{2}{v}^{11}\sin (4\,v) -172169550\,{v}^{11}\sin (2\,v) +712605600\,\cos (v) {s}^{6}{v}^{4}\\&+\, 9525600\,\cos (v) {s}^{4}{v}^{6}+15876000\,\cos (v) {s}^{2}{v}^{8}\\&+\, 201625200\,\sin (v) {s}^{8}v-100699200\,\sin (v) {s}^{6}{v}^{3}\\&-\, 200037600\,\sin (v) {s}^{4}{v}^{5}-152409600\,\cos (v) {s}^{6}{v}^{2}\\&-\, 50058\,\sin (v) {v}^{17}+2467\,{v}^{20}\\&+\, 66868200\,{s}^{8}{v}^{4}\cos (2\,v) -32130\,{v}^{15}\sin (4\,v) \\&-\, 650160\,{s}^{2}{v}^{12}\cos (3\,v) +2425140\,{v}^{15}\sin (2\,v) \\&+\, 5598180\,{s}^{4}{v}^{9}\sin (3\,v) +5238\,{s}^{8}{v}^{8}\cos (3\,v) \\&-\, 10\,{v}^{19}\sin (4\,v) +25924\,{s}^{8}{v}^{10}\cos (2\,v) \\&+\, 3020\,{v}^{19}\sin (2\,v) -188470800\,{s}^{6}{v}^{6}\cos (2\,v) \\&-\, 224916\,{s}^{2}{v}^{15}\sin (2\,v) +340540200\,\cos (v) {s}^{2}{v}^{10}\\&-\, 63997290\,\cos (v) {v}^{12}+275562000\,\cos (v) {v}^{10}\\&-\, 38102400\,\cos (v) {v}^{8}-130410000\,\sin (v) {v}^{9}\\&+\, 195284250\,\sin (v) {v}^{11}-314280\,{s}^{6}{v}^{8}\cos (4\,v) \\&+\, 70345485\,{s}^{8}{v}^{4}-101448\,{s}^{6}{v}^{12}\cos (2\,v) \\&-\, 3702780\,\sin (v) {v}^{15}+76204800\,{s}^{2}{v}^{8}\cos (2\,v) \\&+\, 4934\,{v}^{14}{s}^{6}-1295595\,{v}^{12}\cos (4\,v) \\&-\, 3475080\,{s}^{2}{v}^{11}\sin (3\,v) -6216\,{s}^{2}{v}^{15}\sin (3\,v) \\&+\, 160806\,\sin (v) {s}^{8}{v}^{9}+10666\,\cos (v) {s}^{8}{v}^{10}\\&-\, 65784\,\cos (v) {s}^{6}{v}^{12}-19896840\,{s}^{6}{v}^{8}\cos (2\,v) \\&-\, 197429400\,\sin (v) {s}^{6}{v}^{5}-28533960\,\sin (v) {s}^{2}{v}^{11}\\&-\, 7638120\,{s}^{2}{v}^{13}\sin (2\,v) -1088640\,{s}^{4}{v}^{10}\cos (3\,v) \\&+\, 24403680\,{s}^{2}{v}^{12}-3431970\,{s}^{2}{v}^{14}\\&+\, 428010\,{s}^{6}{v}^{12}-280098000\,{s}^{6}{v}^{4}\\&+\, 56756700\,{s}^{4}{v}^{7}\sin (3\,v) -51672600\,{s}^{2}{v}^{9}\sin (3\,v) \\&+\, 110408\,{s}^{2}{v}^{16}\cos (2\,v) +2\,{v}^{12}{s}^{8}\cos (3\,v) \\&-\, 52920\,{s}^{8}{v}^{6}\cos (3\,v) -1336\,{v}^{14}{s}^{6}\cos (2\,v) \\&+\, 2343600\,{s}^{8}{v}^{2}\cos (4\,v) +278487720\,\cos (v) {s}^{6}{v}^{6}-485408700\,\cos (v) {s}^{4}{v}^{8}\\&+\, 56\,{s}^{2}{v}^{16}\cos (3\,v) -9525600\,{s}^{4}{v}^{6}\cos (3\,v) \\&+\, 28433160\,{s}^{8}{v}^{6}-11592\,\sin (v) {s}^{2}{v}^{15}\\&-\, 15816\,\cos (v) {s}^{6}{v}^{10}-9752400\,{v}^{9}\sin (3\,v) \\&+\, 2\,\cos (3\,v) {v}^{20}-16443000\,{s}^{6}{v}^{5}\sin (3\,v) \\&+\, 63336\,{s}^{6}{v}^{10}\cos (3\,v) -11113200\,{s}^{2}{v}^{8}\cos (4\,v) \\&+\, 53326350\,{s}^{8}{v}^{3}\sin (2\,v) +122301\,{v}^{18}\\&-\, 1159407\,{v}^{16}-14453640\,{v}^{14}-100018800\,{v}^{10}\\&-\, 12879405\,{v}^{12}-194140800\,{v}^{10}\cos (2\,v) \\&+\, 19051200\,{v}^{8}+78\,{v}^{18}\cos (3\,v) \\&-\, 4602\,{s}^{8}{v}^{9}\sin (3\,v) -1851480\,{s}^{6}{v}^{10}\cos (2\,v) \\&-\, 199920\,{s}^{2}{v}^{13}\sin (3\,v) -61020\,{v}^{14}\cos (4\,v) \\&+\, 23110920\,{s}^{2}{v}^{12}\cos (2\,v) -2\,{v}^{18}{s}^{2}\cos (4\,v) \\&+\, 3214890\,{v}^{12}\cos (3\,v) -513000\,{s}^{6}{v}^{7}\sin (3\,v) \\&+\, 2398680\,{s}^{2}{v}^{14}\cos (2\,v) -79380000\,{s}^{8}v\sin (2\,v) \\&-\, 412980\,{s}^{8}{v}^{8}\cos (2\,v) +30660\,{s}^{2}{v}^{13}\sin (4\,v) \\&-\, 270318\,\cos (v) {s}^{8}{v}^{8}-1212\,{s}^{4}{v}^{14}\cos (3\,v) \\&+\, 5940\,{v}^{15}\sin (3\,v) -10615320\,{s}^{6}{v}^{9}\sin (2\,v) \\&+\, 284760\,{s}^{4}{v}^{11}\sin (3\,v) +24844050\,{s}^{2}{v}^{9}\sin (4\,v) \\&+\, 384966\,{s}^{8}{v}^{9}\sin (2\,v) -{v}^{12}{s}^{8}\cos (4\,v) \\&-\, 57153600\,{s}^{2}{v}^{7}\sin (3\,v) +10904220\,{v}^{14}\cos (2\,v) \\&-\, 33339600\,{s}^{8}v\sin (3\,v) -464297400\,\sin (v) {s}^{2}{v}^{9}\\&-\, 35002800\,\cos (v) {s}^{8}{v}^{2}+10001880\,{s}^{6}{v}^{6}\cos (3\,v) \\&+\, 32794740\,{s}^{6}{v}^{7}\sin (2\,v) -19051200\,{v}^{8}\cos (4\,v) \\&-\, 56700\,{s}^{4}{v}^{12}\cos (3\,v) -20326950\,{s}^{2}{v}^{10}\\&-\, 2069550\,{s}^{8}{v}^{3}\sin (3\,v) +3728025\,{s}^{8}{v}^{3}\sin (4\,v) \\&+\, 1944\,{s}^{6}{v}^{12}\cos (3\,v) +66679200\,{s}^{4}{v}^{5}\sin (3\,v) \\&-\, 4934\,{v}^{18}{s}^{2}+71215200\,{s}^{6}{v}^{4}\cos (3\,v) \\&-\, 866\,{s}^{8}{v}^{10}\cos (3\,v) +248535\,{s}^{8}{v}^{5}\sin (4\,v) \\&+\, 152409600\,{s}^{6}{v}^{2}\cos (3\,v) +358\,\cos (v) {s}^{8}{v}^{12}\\&+\, 10073700\,{s}^{8}{v}^{6}\cos (2\,v) +63504000\,{s}^{8}{v}^{2}\cos (2\,v) \\&+\, 8172258\,{s}^{6}{v}^{10}-13396320\,{s}^{6}{v}^{8}\\&-\, 8\,{v}^{18}{s}^{2}\cos (3\,v) -468115200\,{s}^{6}{v}^{4}\cos (2\,v) \\&+\, 38102400\,{v}^{8}\cos (3\,v) -153090\,{s}^{8}{v}^{5}\sin (3\,v) \\&+\, 79740\,{s}^{6}{v}^{9}\sin (4\,v) +12\,{s}^{4}{v}^{16}\cos (3\,v) \\&-\, 9922500\,{s}^{4}{v}^{8}\cos (3\,v) +1285515\,{s}^{8}{v}^{4}\cos (4\,v) \\&-\, 8\,{v}^{14}{s}^{6}\cos (3\,v) +525420\,\cos (v) {s}^{4}{v}^{12}\\&+\, 323640\,\cos (v) {s}^{2}{v}^{14}+100699200\,{s}^{6}{v}^{3}\sin (2\,v) \\&+\, 28\,{s}^{6}{v}^{13}\sin (4\,v) -347178\,{s}^{2}{v}^{16}\\&-\, 12700800\,\cos (v) {s}^{8}-630636\,{s}^{6}{v}^{11}\sin (2\,v) \\&-\, 22\,{s}^{8}{v}^{11}\sin (4\,v) +5509350\,{s}^{8}{v}^{5}\sin (2\,v) \\&+\, 259969500\,{s}^{2}{v}^{9}\sin (2\,v) -24494400\,{s}^{6}{v}^{3}\sin (3\,v) \\&+\, 678\,{s}^{2}{v}^{15}\sin (4\,v) -5400\,{s}^{2}{v}^{14}\cos (3\,v) \\&-\, 15800400\,{s}^{8}{v}^{2}\cos (3\,v) -15044400\,{s}^{8}{v}^{2}\\&+\, 7482\,\cos (v) {v}^{18}-2467\,{v}^{12}{s}^{8}\\&+\, 63\,{v}^{18}\cos (4\,v) +80230500\,{s}^{6}{v}^{5}\sin (2\,v) \\&-\, 334660410\,{s}^{6}{v}^{6}+726\,{v}^{17}\sin (3\,v) \\&+\, 14288400\,{s}^{8}v\sin (4\,v) +28576800\,{s}^{2}{v}^{7}\sin (4\,v) \\&-\, 270\,{s}^{2}{v}^{16}\cos (4\,v) +2\,{v}^{14}{s}^{6}\cos (4\,v) \\&-\, 15750\,{s}^{2}{v}^{14}\cos (4\,v) +5046\,{v}^{16}\cos (3\,v) \\&+\, 129\,{s}^{8}{v}^{10}\cos (4\,v) -297183600\,{s}^{2}{v}^{10}\cos (2\,v) \\&-\, 567000\,{s}^{2}{v}^{12}\cos (4\,v) -65091600\,{s}^{2}{v}^{8}\\&-\, 17917200\,{v}^{10}\cos (4\,v) +93895200\,{v}^{9}\sin (2\,v) \\&-\, 59416\,\cos (v) {s}^{2}{v}^{16}-529320\,\sin (v) {s}^{6}{v}^{11}\\&+\, 430164\,\sin (v) {s}^{4}{v}^{13}-1966230\,{s}^{8}{v}^{4}\cos (3\,v) \\&+\, 107052\,\cos (v) {s}^{4}{v}^{14}-76204800\,{s}^{6}{v}^{2}\cos (4\,v) \\&+\, 264\,{s}^{4}{v}^{15}\sin (3\,v) -135705780\,{s}^{2}{v}^{11}\sin (2\,v) \\&+\, 600\,{s}^{6}{v}^{11}\sin (3\,v) -88236540\,\sin (v) {s}^{4}{v}^{9}\\&+\, 2778\,{s}^{6}{v}^{11}\sin (4\,v) +171460800\,\sin (v) {s}^{2}{v}^{7}\\&+\, 76204800\,{s}^{6}{v}^{2}-203133\,{s}^{8}{v}^{10}\\&+\, 563760\,{s}^{6}{v}^{8}\cos (3\,v) -20460\,{s}^{8}{v}^{7}\sin (3\,v) \\&-\, 130182570\,\cos (v) {s}^{8}{v}^{4}+358\,\cos (v) {v}^{20}\\&-\, 224\,{s}^{6}{v}^{13}\sin (3\,v) +144720\,{s}^{6}{v}^{9}\sin (3\,v) \\&-\, 668\,{v}^{20}\cos (2\,v) -128\,{s}^{2}{v}^{17}\sin (3\,v) \\&+\, 48271230\,\sin (v) {v}^{13}-3535920\,\cos (v) {v}^{14}\\&-\, 6804000\,{s}^{6}{v}^{3}\sin (4\,v) +{v}^{20}\cos (4\,v) \\&-\, 15876000\,{s}^{2}{v}^{8}\cos (3\,v) -13343400\,{s}^{2}{v}^{10}\cos (3\,v) \\&+\, 1336\,{v}^{18}{s}^{2}\cos (2\,v) -344046\,\cos (v) {v}^{16}\\&+\, 20\,\sin (3\,v) {v}^{19}+4\,{s}^{2}{v}^{17}\sin (4\,v) \\&+\, 553675500\,\sin (v) {s}^{4}{v}^{7}+68\,{s}^{8}{v}^{11}\sin (3\,v) \\&-\, 7030800\,{v}^{9}\sin (4\,v) +1140\,{v}^{19}\sin (v) \\&-\, 2987601\,{s}^{8}{v}^{8}+9492\,{s}^{4}{v}^{13}\sin (3\,v) \\&-\, 12768\,\sin (v) {s}^{6}{v}^{13}+15048\,\sin (v) {s}^{4}{v}^{15}\\&-\, 7296\,\sin (v) {s}^{2}{v}^{17}-109005750\,\sin (v) {s}^{8}{v}^{3}\\&+\, 2058060\,{s}^{8}{v}^{7}\sin (2\,v) +441780\,{v}^{16}\cos (2\,v) \\&+\, 14226030\,\sin (v){s}^{8}{v}^{5}+2370\,{s}^{8}{v}^{7}\sin (4\,v) \end{aligned}$$

where $PLM_{j}\left( s,v \right) , \, \, j=8,9,10$ are the Polynomials of the general Stability Functions of the New Introduced Scheme.

Stability District of the New Introduced Scheme $\, \equiv \,$ $s-v$. Figure 5$\longrightarrow$ $s-v$ range of Stability District of the New Introduced Scheme.

Conclusion 4

Figure 6 for $s-v$ $\longrightarrow$:

Colored district of $s-v$ $\longrightarrow$ Stabilized district.
White district of $s-v$ $\longrightarrow$ Unstabilized district.

Conclusion 5

Report on the new nominated Method $\longrightarrow$:

$s \ne v$ $\longrightarrow$ stabilized for $s, \, v$ $\in$ $s-v$ $\backslash$ $\{$ around first diagonal $\}$.
$s = v$ $\longrightarrow$ stabilized for $s, \, v$ $\in$ $\{$ around first diagonal $\}$.

Conclusion 6

Figure 6. $\longrightarrow$ the new deployed procedure is P-stable$\, \equiv \, \langle s-v \rangle \equiv \left( 0, \infty \right)$.

The Table 1$\longrightarrow$ intervals of periodicity of Methods of (4).

Table 1 Intervals of periodicity of the family (4)

Full size table

(4) $+$ (11) $+$ Theorem 2$+$ Stage 5 of Fig. 1$+$ Table 1$\Longrightarrow$

Theorem 3

The new deployed Method:

has 4 levels,
Algebraic Order $\longrightarrow$ 14,
$PhL = 0$,
$DPL_{1}=DPL_{2}=DPL_{3}=DPL_{4}=DPL_{3}=DPL_{5}=DPL_{6}=0$,
is P-stable.

where $PhL = {\text{phase}}{-}{\text{lag}}$ and $DPL_{j} = {\text{derivative of the phase--lag of order j}}$

4 Numerical tests

Figure 7$\longrightarrow$ Valuing the newly deployed Algorithm.

4.1 Local error control procedure

Local Error Control Procedure $\longrightarrow$ Local Truncation Error’s Control Procedure.

Local Error Control Procedure ${\mathop {\longrightarrow }\limits ^{includes}}$: variable–step process (see in [75, 143] and references therein).

Table 2$\longrightarrow$ Papers on the Subject (see [1]–[143]).

Table 2 Literature. Notes: (1) Multistage: Runge–Kutta and of Runge–Kutta Nyström type, P-Stability etc. Involves also multistage schemes with minimal phase–lag. (2) Multistage Exponential: exponentially fitted and trigonometrically fitted Runge–Kutta and Runge–Kutta Nyström Methods. (3) Multistep phase–fitting: Multistep Methods with obliterating phase-lag, obliterating derivatives of the phase–lag, multistep Methods with minimal phase–lag

Full size table

Note 1 Local Error Control Procedure $\longrightarrow$ Procedure for the Control of the Local Truncation Error ${\mathop {\longrightarrow }\limits ^{is \, defined\, as}}$:

Definition 1

$$\begin{aligned} Local Error Control Procedure=\mid \{\beta \lambda \}^{H}_{n+1} - \{\beta \lambda \}^{L}_{n+1} \mid \end{aligned}$$

(32)

where $\{\beta \lambda \}^{L}_{n+1}$ $\longrightarrow$ [86], L $\longrightarrow$ low algebraic order , $\{\beta \lambda \}^{H}_{n+1}$ $\longrightarrow$ Sect. 2, H $\longrightarrow$ high algebraic order .

In [75], the flowchart for Local Error Control Procedure is displayed.

4.2 Description of the problem

$$\begin{aligned} \{\varvec{\beta \lambda }\}'' = - \varvec{\Psi } \, \{\varvec{\beta \lambda }\} \end{aligned}$$

(33)

and

$$\begin{aligned} \Psi _{ij} = \frac{l_{i} (l_{i}+1)}{x^{2}}\, \psi _{ij} + Potent_{ij} \end{aligned}$$

(34)

and $Potent_{ij} \rightarrow 0$ as $x \rightarrow \infty$.

Boundary conditions${\mathop {\longrightarrow }\limits ^{open \, channels}}$

$$\begin{aligned} \{\beta \lambda \}_{ij} = 0 \hbox { } at \hbox { } x = 0 \end{aligned}$$

(35)

$$\begin{aligned} \{\beta \lambda \}_{ij} \sim k_{i} x j_{l_{i}}(k_{i} x) \psi _{ij} + \left( \frac{k_{i}}{k_{j}} \right) ^{1/2} K_{ij} k_{i} x n_{l{i}}(k_{i} x) \end{aligned}$$

(36)

where

$j_{l}(x)$ $\longrightarrow$ spherical Bessel functions and
$n_{l}(x)$ $\longrightarrow$ spherical Neumann functions.

[62] $\longrightarrow$:

$$\begin{aligned}&\frac{2 \nu }{\hbar ^{2}}=1000.0, \hbox { } \frac{\nu }{I}=2.351, \hbox { } E=1.1, \\&V_{0}(x)=\frac{1}{x^{12}}-2 \frac{1}{x^{6}}, \hbox { } V_{2}(x)=0.2283 V_{0}(x). \end{aligned}$$

Figure 8$\longrightarrow$ Flowchart for the problem’s (33) - (35) solution.

Table 3 Computing time (in s) (CT $=$ real time of computation) and maximum absolute error (MBAST). Parameter acurac is needed for we define acurac=10$^{-6}$. $h_length_maximum$ $\longrightarrow$ maximum stepsize. M $\longrightarrow$ equation’s number

Full size table

We use the following schemes:

Method 1: Allison’s Algorithm [62],
Method 2: See [61],
Method 3: See [54],
Method 4: See [65],
Method 5: See [67],
Method 6: See [73],
Method 7: See [49],
Method 8: See [71],
Method 9: See [74],
Method 10: See [37],
Method 11: See [85],
Method 12: See [40],
Method 13: See [41],
Method 14: See [42],
Method 15: See [43],
Method 16: See [44],
Method 17: Sect. 2.

Table 3$\longrightarrow$ CT (in seconds) and MBAST for the 16 Methods.

All computations were carried out on a x86-64 compatible PC using double-precision arithmetic data type (64 bits) according to IEEE^© Standard 754 for double precision.

5 General conclusion

A new FD method with phase–lag and its derivatives up to order 6 equal to zero is protduced. The theoretical evaluation which is presented in Sect. 3, proves the theoretical superiority of the new introduced scheme. The computational evaluation of the new method, which is based on (33), proves the efficiency of the new scheme.

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Ural Federal University, Mira 19, Yekaterinburg, Russian Federation
Maxim A. Medvedev
Institute of Industrial Ecology UB RAS, Kovalevskaya St. 20, Yekaterinburg, Russian Federation
Maxim A. Medvedev
South Ural State University, 76, Lenin Aven., Chelyabinsk, Russian Federation, 454080
T. E. Simos
Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Dongtong Road 705, Neijiang, 641100, China
T. E. Simos
Group of Modern Computational Methods, Ural Federal University, Mira 19, Yekaterinburg, Russian Federation
T. E. Simos
Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece
T. E. Simos
10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64, Athens, Greece
T. E. Simos

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Medvedev, M.A., Simos, T.E. New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry. J Math Chem 58, 2324–2360 (2020). https://doi.org/10.1007/s10910-020-01168-5

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Received: 13 August 2020
Accepted: 04 September 2020
Published: 29 September 2020
Issue Date: November 2020
DOI: https://doi.org/10.1007/s10910-020-01168-5

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New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry

Abstract

1 Introduction

1.1 The problem

2 Introductiuon of the algorithm

Theorem 1

3 Theoretical study of the new proposed method

3.1 Theoretical study of the local truncation error of the new proposed method

3.1.1 Classical algorithm, related as method A

3.1.2 Procedure projected in [40], related as method B

3.1.3 Procedure projected in [41], related as method C

3.1.4 Procedure projected in [42], related as Method D

3.1.5 Procedure projected in [43], related as Method E

3.1.6 Procedure projected in in [44], Related as Method F

3.1.7 Procedure projected in Section 2, Related as Method G

Conclusion 1

Conclusion 2

3.1.8 Method A

3.1.9 Method B

3.1.10 Method C

3.1.11 Method D

3.1.12 Method E

3.1.13 Method F

3.1.14 Method G

Theorem 2

3.2 Theoretical study of the stability (stability analysis of the new introduced method

Conclusion 3

Conclusion 4

Conclusion 5

Conclusion 6

Theorem 3

4 Numerical tests

4.1 Local error control procedure

Definition 1

4.2 Description of the problem

5 General conclusion

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