1 Introduction
2 Conditions for symmetry, symplecticity, exponential fitting of modified RKN methods
c
| e γ | A |
1 1 |
\(\bar{b}^{T}\)
| |
1 |
\(b^{T}\)
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\(c_{1}\)
| 1 |
\(\gamma_{1}\)
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\(a_{11}\)
| ⋯ |
\(a_{1s}\)
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⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
\(c_{s}\)
| 1 |
\(\gamma_{s}\)
|
\(a_{s1}\)
| ⋯ |
\(a_{ss}\)
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1 | 1 |
\(\bar{b}_{1}\)
| ⋯ |
\(\bar {b}_{s}\)
| |
1 |
\(b_{1}\)
| ⋯ |
\(b_{s}\)
|
2.1 Symmetry conditions
2.2 Symplecticity conditions
2.3 Exponential fitting conditions
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for the internal stages,$$ \varphi_{i}\bigl[y(t);h;\mathbf{a}\bigr]=y(t+c_{i}h)-y(t)-c_{i} \gamma _{i}hy'(t)-h^{2}\sum _{j=1}^{s}a_{ij}y''(t+c_{j}h), \quad i=1,2,\ldots,s; $$
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for the final stages,$$ \textstyle\begin{cases} \varphi[y(t);h;\bar{\mathbf{b}}]=y(t+h)-y(t)-hy'(t)-h^{2}\sum_{i=1}^{s}\bar{b}_{i}y''(t+c_{i}h), \\ \varphi[y(t);h;\mathbf{b}]=y'(t+h)-y'(t)-h\sum_{i=1}^{s}b_{i}y''(t+c_{i}h). \end{cases} $$
3 Algebraic order conditions
4 Construction of implicit symmetric and symplectic modified EFRKN methods
5 Numerical experiments
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DIRKNRaed: The embedded diagonally implicit RKN 4(3) pair method proposed by Al-Khasawneh et al. in [1].
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DIRKNNora: The three-stage fourth-order diagonally implicit RKN method proposed by Senu et al. in [27].
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ISSRKN2: The symmetric and symplectic two-stage fourth-order implicit RKN method proposed by MENG-ZHAO QIN et al. in [24] with \(a_{11}=\frac{-13+160\theta^{2}+720\theta^{4}}{2880\theta^{2}}\) and \(\theta=\pm\frac{\sqrt{3}}{6}\), i.e., \(a_{11}=\frac{1}{45}\).
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ISSEFRKN2: The symmetric and symplectic exponentially fitted two-stage RKN method proposed in [35] which is of order 4.
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ISSEFMRKN2: The symmetric and symplectic exponentially fitted two-stage modified RKN method (25) proposed in this paper which is of order 4.