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Algorithm for the development of families of numerical methods based on phase-lag Taylor series

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Abstract

In the present research paper we propose a new generator of families of numerical methods with increasing number of internal layers in an attempt to achieve higher order accuracy. The intermediate stages consist of predictor-corrector methods. The final layer is a symmetric two step method with constant coefficients and also free parameters. Those parameters define each family of methods. At first the method is constructed with unknown parameters and subsequently their value is estimated in order to fulfill the requirement of maximum phase-lag order. The stability of the new numerical algorithm is analyzed and the local truncation error is computed. The generator of the new families is applied to well known problems and is found to be more efficient compared to other methods and numerical methods generators with similar characteristics, which attempt to numerically solve such problems.

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Abbreviations

LTE:

Local Truncation Error

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Acknowledgements

This work was supported by Special Account for Research Funds(SARF) of Democritus University of Thrace-project no. 81896.

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  1. Department of Civil Engineering, Polytechnic School of Xanthi, Democritus University of Thrace, Campus Xanthi, Kimmeria, 67100, Xanthi, Greece

    A. Konguetsof

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Appendix

Appendix

1.1 Phase-Lag Formula Confirmation

The phase-lag formula for \(k=1\) is obtained

$$\begin{aligned} PhL_1=T_0 + T_1 d_1=T_0+(A_1 v^6 + B_1 v^8) d_1 \end{aligned}$$
(66)

and for the value \(k=2\) takes the form

$$\begin{aligned} PhL_2= & {} T_0 + T_1 \prod _{j=2}^{2} d_j + T_2 \prod _{j=1}^{2} d_j \nonumber \\= & {} T_0+(A_1 v^6 + B_1 v^8) d_2 +(A_2 v^8 + B_2 v^{10}) d_1 d_2 \end{aligned}$$
(67)

The phase-lag formula for \(k=n\) and \(i=1, 2...,n\) is as follows

$$\begin{aligned} PhL_n=T_0+\sum _{i=1}^{n} \left( T_i \prod _{j=n+1-i}^{n} d_j \right) \end{aligned}$$
(68)

The phase-lag formula for \(k=n+1\) and consequently \(i=1, 2...,n+1\) is confirmed by the previous mathematical representations (66), (67) and (68)

$$\begin{aligned} PhL_{n+1}= & {} T_0+\sum _{i=1}^{n+1} \left( T_i \prod _{j=n+2-i}^{n+1} d_j \right) \nonumber \\= & {} T_0 + \sum _{i=1}^{n} \left( T_i \prod _{j=n+2-i}^{n+1} d_j \right) + T_{n+1} \prod _{j=1}^{n+1} d_j, \end{aligned}$$
(69)

that can be easily verified after expanding the sum and the product in an analytical way.

1.2 Confirmation of general formula for the system of equation for the solution of the unknown parameters

For a given family of methods, so that the value of k remains unchanged and the value of \(m \in {\mathbb {N}}\) with \(m \ge 3\) is variable, the following procedure describes our general expressions (31) given.

If \(m=3\) we obtain the equation

$$\begin{aligned} F^k(3)=-\frac{5}{3} d_{k-2} d_{k-1} d_{k} - \frac{5}{72} d_{k-1} d_{k} -\frac{1}{B_F}, \end{aligned}$$
(70)

where \(B_F=3628800\).

For \(m=n\) the expression obtained is

$$\begin{aligned}&F^k(n)=\frac{(-1)^{n}}{3} \frac{5}{ 2^{3-n} } \prod _{j=k-n+1}^{k} d_j \nonumber \\&+\frac{(-1)^{n}}{9} \frac{5}{ 2^{6-n} } \prod _{j=k-n+2}^{k} d_j + \frac{(-1)^n}{{B_F {\tilde{F}}^k_j(n)}} \nonumber \\&=\frac{(-1)^{n}}{3} \frac{5}{2^{3-n} } d_{k-n+1} d_{k-n}... d_{k-1} d_{k} \nonumber \\&+\frac{(-1)^{n}}{9} \frac{5}{ 2^{6-n} } d_{k-n+2} d_{k-n+1}... d_{k-1} d_{k}+ \frac{(-1)^n}{B_F {\tilde{F}}^k_j(n)}=0 , \end{aligned}$$
(71)

where \({\tilde{F}}^k_j(n)=\prod _{j=1}^{n-3} \left( 90+38j+4 j^2 \right) \).

For \(m=n+1\) the above expression takes the form

$$\begin{aligned}&F^k(n+1)=\frac{(-1)^{n+1}}{3} \frac{5}{ 2^{2-n} } \prod _{j=k-n}^{k} d_j \nonumber \\&+\frac{(-1)^{n+1}}{9} \frac{5}{ 2^{5-n} } \prod _{j=k-n+1}^{k} d_j + \frac{(-1)^{n+1}}{{B_F {\tilde{F}}^k_j(n+1)}} \nonumber \\&=\frac{(-1)^{n+1}}{3} \frac{5}{2^{2-n} } d_{k-n}... d_{k-1} d_{k} \nonumber \\&+\frac{(-1)^{n+1}}{9} \frac{5}{ 2^{5-n} } d_{k-n+1} ... d_{k-1} d_{k}+ \frac{(-1)^{n+1}}{B_F {\tilde{F}}^k_j(n+1)}=0 , \end{aligned}$$
(72)

where \({\tilde{F}}^k_j(n+1)=\prod _{j=1}^{n-2} \left( 90+38j+4 j^2 \right) \).

If \({\tilde{F}}^k_j(3)=\prod _{j=1}^{m-3} \left( 90+38j+4 j^2 \right) \), then for \(m=3\), we have \({\tilde{F}}^k_j(3)=\prod _{j=1}^{0} \left( 90+38j+4 j^2 \right) \), which is considered equal to one.

Respectively for \(m=n\) the quantity \(F^k_j(n)\) is

$$\begin{aligned} {\tilde{F}}^k_j(n)=\prod _{j=1}^{n-3} s^k(j) \end{aligned}$$
(73)

For \(m=n+1\) it becomes as follows

$$\begin{aligned} {\tilde{F}}^k_j(n+1)=\prod _{j=1}^{n-2} s^k(j), \end{aligned}$$
(74)

with \(s^k(j)=90+38j+4j^2\).

So that the general formula, which forms the system of equations (29)-(31) that leads to the values of the intermediate parameters, is verified.

1.3 Local Truncation Error for some families of the generator

The local truncation error \(LER_k\) is computed for the families with \(k=4,...,10\) and \(k=13\).

$$\begin{aligned} LER_4= & {} - {{7393} \over {32691859200}}\left( {{{{d^{14}}} \over {d{t^{14}}}}y\left( t \right) } \right) {h^{14}} \nonumber \\&+ {1 \over {10461394944000}}\left( {{{{d^{16}}} \over {d{t^{16}}}}y\left( t \right) } \right) {h^{16}} \nonumber \\&+ {1 \over {3201186852864000}}\left( {{{{d^{18}}} \over {d{t^{18}}}}y\left( t \right) } \right) {h^{18}} \nonumber \\&+{1 \over {1216451004088320000}}{\left( {{{{d^{20}}} \over {d{t^{20}}}}y\left( t \right) } \right) }{h^{20}} +... , \end{aligned}$$
(75)
$$\begin{aligned} LER_5= & {} {{{{591443}}} \over {{{31384184832000}}}}{\left( {{{{d^{16}}} \over {d{t^{16}}}}y\left( t \right) } \right) }{h^{16}} \nonumber \\&+ {1 \over {{{3201186852864000}}}}{\left( {{{{d^{18}}} \over {d{t^{18}}}}y\left( t \right) } \right) }{h^{18}} \nonumber \\&+ {1 \over {{ {1216451004088320000}}}}{\left( {{{{d^{20}}} \over {d{t^{20}}}}y\left( t \right) } \right) }{h^{20}} +... , \end{aligned}$$
(76)
$$\begin{aligned} LER_6= & {} - {{{ {10054529}}} \over {{ {6402373705728000}}}}{\left( {{{{d^{18}}} \over {d{t^{18}}}}y\left( t \right) } \right) }{h^{18}} \nonumber \\&+ {1 \over {{ {1216451004088320000}}}}{\left( {{{{d^{20}}} \over {d{t^{20}}}}y\left( t \right) } \right) }{h^{20}} +... , \end{aligned}$$
(77)
$$\begin{aligned} LER_7= & {} {{{ {136454323}}} \over {{ {1042672289218560000}}}}\left( {{{{d^{20}}} \over {d{t^{20}}}}y\left( t \right) } \right) {h^{20}}+... , \end{aligned}$$
(78)
$$\begin{aligned} LER_8= & {} - {{{ {136454323}}} \over {{ {12512067470622720000}}}}{\left( {{{{d^{22}}} \over {d{t^{22}}}}y\left( t \right) } \right) }{h^{22}}+... , \end{aligned}$$
(79)
$$\begin{aligned} LER_9= & {} - {1 \over {{ {562000363888803840000}}}}\left( {{{{d^{22}}} \over {d{t^{22}}}}y\left( t \right) } \right) {h^{22}} \nonumber \\&+ {{{ {14709776017}}} \over {{ {16185610479997550592000}}}}\left( {{{{d^{24}}} \over {d{t^{24}}}}y\left( t \right) } \right) {h^{24}}+... , \end{aligned}$$
(80)
$$\begin{aligned} LER_{10}= & {} - {1 \over {{ {562000363888803840000}}}}\left( {{{{d^{22}}} \over {d{t^{22}}}}y\left( t \right) } \right) {h^{22}} \nonumber \\&- {1 \over {{ {310224200866619719680000}}}}\left( {{{{d^{24}}} \over {d{t^{24}}}}y\left( t \right) } \right) {h^{24}} \nonumber \\&- {{{ {1691624241961}}} \over {{ {22336142462396619816960000}}}}\left( {{{{d^{26}}} \over {d{t^{26}}}}y\left( t \right) } \right) {h^{26}} +... , \end{aligned}$$
(81)
$$\begin{aligned} LER_{13}= & {} - {1 \over {{ {562000363888803840000I}}}} \left( {{{{d^{22}}} \over {d{t^{22}}}}y\left( t \right) } \right) {h^{22}} \nonumber \\&- {1 \over {{ {310224200866619719680000}}}} \left( {{{{d^{24}}} \over {d{t^{24}}}}y\left( t \right) } \right) {h^{24}} \nonumber \\&- {1 \over {20{ {1645730563302817792000000}}}} \left( {{{{d^{26}}} \over {d{t^{26}}}}y\left( t \right) } \right) {h^{26}} \nonumber \\&- {1 \over {{ { 1 52444172305856930250752000000}}}} \left( {{{{d^{28}}} \over {d{t^{28}}}}y\left( t \right) } \right) {h^{28}} \nonumber \\&- {1 \over {{ {132626429906095529318154240000000}}}} \left( {{{{d^{30}}} \over {d{t^{30}}}}y\left( t \right) } \right) {h^{30}} \nonumber \\&+ {{{ {20667575808031441}}} \over {{ {471560639666117437575659520000000}}}} \left( {{{{d^{32}}} \over {d{t^{32}}}}y\left( t \right) } \right) {h^{32}} +... \nonumber \\ \end{aligned}$$
(82)

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Konguetsof, A. Algorithm for the development of families of numerical methods based on phase-lag Taylor series. J Math Chem 58, 178–203 (2020). https://doi.org/10.1007/s10910-019-01078-1

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