A class of linear multi-step method adapted to general oscillatory second-order initial value problems

Abstract

This paper proposes and investigates a class of new linear multi-step methods for solving general oscillatory second-order initial value problems $y^{\prime \prime }(t)+My(t)=f(t,y(t),y^{\prime }(t))$, where M is a positive semi-definite matrix containing implicitly the frequencies of the problem. The new methods, with coefficients depending on the frequency matrix M, incorporate the special structure of the problem brought by the linear term My(t) and integrate exactly the unperturbed oscillator $y^{\prime \prime }(t)+My(t)=\mathbf {0}$. This class of new methods can be viewed extensions of famous Gautschi-type methods from special oscillatory problem to general oscillatory problem. A rigorous error analysis is given and the local truncation errors of the solution and the derivative are presented. Numerical experiments show that our new methods are more efficient in comparison with the well-known high quality methods proposed in the scientific literature.

Appendices

Appendix 1: The proof of (15)

In fact

$$\begin{aligned} \sum \limits _{l=0}^{k-1}\bar{b}_{l+1}(V)f_{n-l}=\sum \limits _{j=0}^{k-1}\sigma _j(V)\nabla ^jf_{n} =\sum \limits _{j=0}^{k-1}\sigma _j(V)\sum \limits _{l=0}^j\left( \begin{array}{c}j\\ l\end{array}\right) (-1)^lf_{n-l} \end{aligned}$$

results in

$$\begin{aligned} \bar{b}_{l+1}(V)=\sum \limits _{j=l}^{k-1}\sigma _j(V)\left( \begin{array}{c}j\\ l\end{array}\right) (-1)^l. \end{aligned}$$

Replacing l with $i-1$, we obtain the first formula of (15). The second one can be obtained similarly.

Appendix 2: Specific examples of particular methods

ASNSMS3K4P: The method is designed for $y^{\prime \prime }+My=f(t,y,y^{\prime })$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&\ \bar{y}_{n+1}=2\phi _0(V)y_n-y_{n-1}+h^2\sum \limits _{j=0}^{2}\sigma _j(V)\nabla ^jf_n,\\&\ \bar{y}_{n+1}^{\prime }=-2hM\phi _1(V)y_n+y_{n-1}^{\prime }+h\sum \limits _{j=0}^{2}\kappa _j(V)\nabla ^jf_n,\\&\ y_{n+1}=\ 2\phi _0(V)y_n-y_{n-1}+h^2\sum \limits _{j=0}^{3}\sigma _j^*(V)\nabla ^jf_{n+1},\\&\ y_{n+1}^{\prime }=\ -2hM\phi _1(V)y_n+y_{n-1}^{\prime }+h\sum \limits _{j=0}^{3}\kappa _j^*(V)\nabla ^jf_{n+1},\\ \end{aligned}\right. \end{aligned}$$

(48)

where $f_{n+1}=f(t_{n+1},\bar{y}_{n+1},\bar{y^{\prime }}_{n+1})$ and $f_{j}=f(t_{j},y_{j},y_{j}^{\prime })$ for $j=n,n-1,n-2$. $\nabla ^jf_n$ denotes the jth backward difference, defined recursively by

$$\begin{aligned} \nabla ^0f_n=f_n,\quad \nabla ^jf_n=\nabla ^{j-1}f_n-\nabla ^{j-1}f_{n-1},\quad j=1,2,3, \end{aligned}$$

and

$$\begin{aligned} \sigma _0(V)&=2\phi _2(V),\quad \sigma _1(V)=\mathbf {0},\quad \sigma _2(V)=2\phi _4(V),\\ \kappa _0(V)&=2\phi _1(V),\quad \kappa _1(V)=\mathbf {0},\quad \kappa _2(V)=2\phi _3(V),\\ \sigma _0^*(V)&=2\phi _2(V),\quad \sigma _1^*(V)=-2\phi _2(V),\quad \sigma _2^*(V)=2\phi _4(V),\quad \sigma _3^*(V)=\mathbf {0},\\ \kappa _0^*(V)&=2\phi _1(V),\quad \kappa _1^*(V)=-2\phi _1(V),\quad \kappa _2^*(V)=2\phi _3(V),\quad \kappa _3^*(V)=\mathbf {0}. \end{aligned}$$

ASNSMS4K5P: The method is designed for $y^{\prime \prime }+My=f(t,y,y^{\prime })$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&\ \bar{y}_{n+1}=2\phi _0(V)y_n-y_{n-1}+h^2\sum \limits _{j=0}^{3}\sigma _j(V)\nabla ^jf_n,\\&\ \bar{y}_{n+1}^{\prime }=-2hM\phi _1(V)y_n+y_{n-1}^{\prime }+h\sum \limits _{j=0}^{3}\kappa _j(V)\nabla ^jf_n,\\&\ y_{n+1}=\ 2\phi _0(V)y_n-y_{n-1}+h^2\sum \limits _{j=0}^{4}\sigma _j^*(V)\nabla ^jf_{n+1},\\&\ y_{n+1}^{\prime }=\ -2hM\phi _1(V)y_n+y_{n-1}^{\prime }+h\sum \limits _{j=0}^{4}\kappa _j^*(V)\nabla ^jf_{n+1},\\ \end{aligned}\right. \end{aligned}$$

(49)

where $f_{n+1}=f(t_{n+1},\bar{y}_{n+1},\bar{y^{\prime }}_{n+1})$ and $f_{j}=f(t_{j},y_{j},y_{j}^{\prime })$ for $j=n,n-1,n-2,n-3$. $\nabla ^jf_n$ denotes the jth backward difference, defined recursively by

$$\begin{aligned} \nabla ^0f_n=f_n,\quad \nabla ^jf_n=\nabla ^{j-1}f_n-\nabla ^{j-1}f_{n-1},\quad j=1,2,3,4 \end{aligned}$$

and

$$\begin{aligned} \sigma _0(V)&=2\phi _2(V),\quad \sigma _1(V)=\mathbf {0},\quad \sigma _2(V)=2\phi _4(V),\quad \sigma _3(V)=2\phi _4(V),\\ \kappa _0(V)&=2\phi _1(V),\quad \kappa _1(V)=\mathbf {0},\quad \kappa _2(V)=2\phi _3(V),\quad \kappa _3(V)=2\phi _3(V),\\ \sigma _0^*(V)&=2\phi _2(V),\quad \sigma _1^*(V)=-2\phi _2(V),\quad \sigma _2^*(V)=2\phi _4(V),\quad \sigma _3^*(V)=\mathbf {0},\\ \sigma _4^*(V)&=\frac{1}{6}(-\phi _4(V)+12\phi _6(V)),\\ \kappa _0^*(V)&=2\phi _1(V),\quad \kappa _1^*(V)=-2\phi _1(V),\quad \kappa _2^*(V)=2\phi _3(V),\quad \kappa _3^*(V)=\mathbf {0},\\ \kappa _4^*(V)&=-\frac{1}{6}\phi _3(V)+2\phi _5(V). \end{aligned}$$

ADAMS3K4P:The method is designed for $y^{\prime }=f(t,y)$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&\ \bar{y}_{n+1}=y_n+h\left( \frac{23}{12}f_n-\frac{16}{12}f_{n-1}+\frac{5}{12}f_{n-2}\right) ,\\&\ y_{n+1}=\ y_n+h\left( \frac{9}{24}f_{n+1}+\frac{19}{24}f_n-\frac{5}{24}f_{n-1}+\frac{1}{24}f_{n-2}\right) ,\\ \end{aligned}\right. \end{aligned}$$

where $f_{n+1}=f(t_{n+1},\bar{y}_{n+1})$ and $f_{j}=f(t_{j},y_{j})$ for $j=n,n-1,n-2$.

ADAMS4K5P:The method is designed for $y^{\prime }=f(t,y)$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&\ \bar{y}_{n+1}=y_n+h\left( \frac{55}{24}f_n-\frac{59}{24}f_{n-1}+\frac{37}{24}f_{n-2}-\frac{9}{24}f_{n-3}\right) ,\\&\ y_{n+1}=\ y_n+h\left( \frac{251}{720}f_{n+1}+\frac{646}{720}f_n-\frac{264}{720}f_{n-1}+\frac{106}{720}f_{n-2}-\frac{19}{720}f_{n-3}\right) , \end{aligned}\right. \end{aligned}$$

where $f_{n+1}=f(t_{n+1},\bar{y}_{n+1})$ and $f_{j}=f(t_{j},y_{j})$ for $j=n,n-1,n-2,n-3$.

RKN4S4P: The method is designed for $y^{\prime \prime }=f(t,y,y^{\prime })$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&Y_i=y_n+c_ihy_{n}^{\prime } +h^2\textstyle \sum \limits _{j=1}^{i-1}\bar{a}_{ij}f(t_n+c_jh,Y_j,Y_j^{\prime }),\quad i=1,\ldots ,4, \\&Y_i^{\prime }=y_{n}^{\prime } +h\textstyle \sum \limits _{j=1}^{i-1}a_{ij}f(t_n+c_jh,Y_j,Y_j^{\prime }),\quad i=1,\ldots ,4, \\&y_{n+1}=y_n+hy_{n}^{\prime }+h^2\textstyle \sum \limits _{i=1}^4\bar{b}_if(t_n+c_ih,Y_i,Y_i^{\prime }),\\&y_{n+1}^{\prime }=y_n^{\prime }+h\textstyle \sum \limits _{i=1}^4b_if(t_n+c_ih,Y_i,Y_i^{\prime }), \end{aligned}\right. \end{aligned}$$

where

$$\begin{aligned} c_1&= 0,\quad c_2 = \frac{1}{2},\quad c_3 =\frac{1}{2},\quad c_4 = 1, \\ a_{21}&=\frac{1}{2},\quad a_{31} =0,\quad a_{32 }= \frac{1}{2}, \quad a_{41} = 0, \quad a_{42} = 0,\quad a_{43} = 1, \\ \bar{a}_{21}&= \frac{1}{8},\quad \bar{a}_{31} = \frac{1}{8}, \quad \bar{a}_{32} = 0,\quad \bar{a}_{41} = 0,\quad \bar{a}_{42} = 0 , \quad \bar{a}_{43 }=\frac{1}{2},\\ b_1&= \frac{1}{6}, \quad b_2 = \frac{2}{6},\quad b_3 = \frac{2}{6}, \quad b_4 = \frac{1}{6},\\ \bar{b}_1&= \frac{1}{6}, \quad \bar{b}_2 = \frac{1}{6},\quad \bar{b}_3 = \frac{1}{6}, \quad \bar{b}_4 =0. \end{aligned}$$

RK4S4P:The method is designed for $y^{\prime }=f(t,y)$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&Y_i=y_n+h\textstyle \sum \limits _{j=1}^{i-1}a_{ij}f(t_n+c_jh,Y_j),\quad i=1,\ldots ,4, \\&y_{n+1}=y_n+h\textstyle \sum \limits _{i=1}^4 b_if(t_n+c_ih,Y_i),\\ \end{aligned}\right. \end{aligned}$$

where

$$\begin{aligned}&c_1 = 0,\quad c_2 = \frac{1}{3},\quad c_3 =\frac{2}{3},\quad c_4 = 1, \\&a_{21} =\frac{1}{3},\quad a_{31} =-\frac{1}{3},\quad a_{32 }= 1,\quad a_{41} = 1, \quad a_{42} =-1,\quad a_{43} = 1, \\&b_1 = \frac{1}{8}, \quad b_2 = \frac{3}{8},\quad b_3 = \frac{3}{8},\quad b_4 = \frac{1}{8}. \end{aligned}$$

ARKN4S4P: The method is designed for $y^{\prime \prime }+My=f(t,y,y^{\prime })$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&Y_i=y_n+c_ihy_{n}^{\prime } +h^2\textstyle \sum \limits _{j=1}^{i-1}\bar{a}_{ij}(-MY_j+f(t_n+c_jh,Y_j,Y_j^{\prime })),\quad i=1,\ldots ,4, \\&Y_i^{\prime }=y_{n}^{\prime } +h\textstyle \sum \limits _{j=1}^{i-1}a_{ij}(-MY_j+f(t_n+c_jh,Y_j,Y_j^{\prime })),\quad i=1,\ldots ,4, \\&y_{n+1}=\phi _0(V)y_n+h\phi _1(V)y_{n}^{\prime }+h^2\textstyle \sum \limits _{i=1}^4\bar{b}_if(t_n+c_ih,Y_i,Y_i^{\prime }),\\&y_{n+1}^{\prime }=-hM\phi _1(V)y_{n}^{\prime }+\phi _0(V)y_{n}^{\prime }+h\textstyle \sum \limits _{i=1}^4b_if(t_n+c_ih,Y_i,Y_i^{\prime }),\\ \end{aligned}\right. \end{aligned}$$

where

$$\begin{aligned} c_1&= 0,\quad c_2 = \frac{1}{2},\quad c_3 =\frac{1}{2},\quad c_4 = 1, \\ a_{21}&=\frac{1}{2},\quad a_{31} =0,\quad a_{32 }= \frac{1}{2},\quad a_{41} = 0, \quad a_{42} = 0,\quad a_{43} = 1, \\ \bar{a}_{21}&= 0,\quad \bar{a}_{31} = \frac{1}{4}, \quad \bar{a}_{32} = 0,\quad \bar{a}_{41} = 0,\quad \bar{a}_{42} = \frac{1}{2}, \quad \bar{a}_{43 }= 0, \\ b_1(V)&= \phi _1(V)-3\phi _2(V) + 4\phi _3(V), \quad \quad \bar{b}_1(V) = \phi _2(V)-3\phi _3(V) + 4\phi _4(V),\\ b_2(V)&= 2\phi _2(V)- 4\phi _3(V), \quad \quad \quad \bar{ b}_2(V) = 2\phi _3(V) - 4\phi _4(V), \\ b_3(V)&= 2\phi _2(V)- 4\phi _3(V), \quad \quad \quad \bar{b}_3(V) = 2\phi _3(V) - 4\phi _4(V),\\ b_4(V)&= -\phi _2(V) + 4\phi _3(V), \quad \quad \quad \bar{b}_4(V) = -\phi _3(V) + 4\phi _4(V). \end{aligned}$$

ARKN6S5P: The method is designed for $y^{\prime \prime }+My=f(t,y,y^{\prime })$ and can be expressed as

$$\begin{aligned} \left\{ \begin{aligned}&Y_i=y_n+c_ihy_{n}^{\prime } +h^2\textstyle \sum \limits _{j=1}^{i-1}\bar{a}_{ij}(-MY_j+f(t_n+c_jh,Y_j,Y_j^{\prime })),\quad i=1,\ldots ,6, \\&Y_i^{\prime }=y_{n}^{\prime } +h\textstyle \sum \limits _{j=1}^{i-1}a_{ij}(-MY_j+f(t_n+c_jh,Y_j,Y_j^{\prime })),\quad i=1,\ldots ,6, \\&y_{n+1}=\phi _0(V)y_n+h\phi _1(V)y_{n}^{\prime }+h^2\textstyle \sum \limits _{i=1}^6\bar{b}_if(t_n+c_ih,Y_i,Y_i^{\prime }),\\&y_{n+1}^{\prime }=-hM\phi _1(V)y_{n}^{\prime }+\phi _0(V)y_{n}^{\prime }+h\textstyle \sum \limits _{i=1}^6b_if(t_n+c_ih,Y_i,Y_i^{\prime }),\\ \end{aligned}\right. \end{aligned}$$

where

$$\begin{aligned} c_1&= 0,\quad c_2 = \frac{1}{6},\quad c_3 =\frac{2}{6},\quad c_4 = \frac{3}{6},\quad c_5 =\frac{4}{6},\quad c_6 = 1, \\ a_{21}&=\frac{1}{6},\quad a_{31} =0,\quad a_{32 }= \frac{1}{3},\quad a_{41} = -\frac{1}{4}, \quad a_{42} = \frac{3}{4},\quad a_{43} = 0, \\ a_{51}&=-\frac{1}{27},\quad a_{52} =\frac{2}{9},\quad a_{53 }= \frac{1}{3},\quad a_{54} = \frac{4}{27},\\ a_{61}&=-\frac{2}{11},\quad a_{62} =\frac{3}{11},\quad a_{63 }= \frac{27}{11},\quad a_{64} =-4,\quad a_{65} = \frac{27}{11},\\ \bar{a}_{21}&= 0,\quad \bar{a}_{31} = \frac{1}{18}, \quad \bar{a}_{32} = 0,\quad \bar{a}_{41} = \frac{1}{8},\quad \bar{a}_{42} =0, \quad \bar{a}_{43 }= 0,\\ \bar{a}_{51}&= 0,\quad \bar{a}_{52} = \frac{2}{9}, \quad \bar{a}_{53} = 0,\quad \bar{a}_{54} = 0,\\ \bar{a}_{61}&=\frac{21}{22},\quad \bar{a}_{62} = -\frac{18}{11}, \quad \bar{a}_{63} = \frac{9}{11},\quad \bar{a}_{64} = \frac{4}{11},\quad \bar{a}_{65} = 0,\\ \end{aligned}$$

and

$$\begin{aligned} b_1(V)&= \phi _1(V)-\frac{15}{2}\phi _2(V) + 40\phi _3(V)- 135\phi _4(V)+216\phi _5(V),\quad b_2(V)=\mathbf {0},\\ b_3(V)&= 27(\phi _2(V)- 9\phi _3(V)+ 39\phi _4(V)-72\phi _5(V),\\ b_4(V)&= -32\left( \phi _2(V)- 11\phi _3(V)+ 54\phi _4(V)-108\phi _5(V)\right) ,\\ b_5(V)&= \frac{27}{2}\left( \phi _2(V)- 12\phi _3(V)+ 66\phi _4(V)-144\phi _5(V)\right) ,\\ b_6(V)&= -\phi _2(V)+13\phi _3(V)- 81\phi _4(V)+216\phi _5(V),\\ \bar{b}_1(V)&= \phi _2(V)-5\phi _3(V)+\frac{64}{5}\phi _4(V)-13\phi _5(V),\quad \bar{b}_2(V)=\mathbf {0},\\ \bar{b}_3(V)&=9\phi _3(V)-\frac{171}{5}\phi _4(V)+45\phi _5(V),\\ \bar{b}_4(V)&=-4\phi _3(V)+\frac{64}{5}\phi _4(V)-16\phi _5(V),\\ \bar{b}_5(V)&=\frac{54}{5}\phi _4(V)-27\phi _5(V),\quad \bar{b}_6(V) =-\frac{11}{5}\phi _4(V)+11\phi _5(V),\\ \end{aligned}$$