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Erschienen in:
23.03.2016 | Erratum
Erratum to: Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes
Erschienen in: Journal of Scientific Computing | Ausgabe 3/2016
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Excerpt
The authors regret that typographical errors appeared in the order conditions, Table 1 in the original publication. These errors included a mistaken factor of 2 on one of the terms in one of the fifth-order conditions, and an omitted equation. The corrected Table 1 of order conditions is provided here.
Table 1
Order conditions for multistage multiderivative methods of the form (6) as in [3]
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\(p = 1\)
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\(b^T e =1\)
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\(p = 2\)
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\(b^T c+\hat{b}^Te =\frac{1}{2}\)
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\(p= 3\)
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\(b^T c^2 + 2\hat{b}^T c=\frac{1}{3}\)
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\(b^TAc+b^T\hat{c}+\hat{b}^Tc=\frac{1}{6}\)
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\(p=4\)
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\(b^Tc^3+3\hat{b}^Tc^2=\frac{1}{4}\)
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\(b^TcAc+b^Tc\hat{c}+\hat{b}^Tc^2+\hat{b}^TAc+\hat{b}^T\hat{c}=\frac{1}{8}\)
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\(b^TAc^2+2b^T\hat{A}c+\hat{b}^Tc^2=\frac{1}{12}\)
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\(b^TA^2c+ b^TA\hat{c}+ b^T\hat{A}c+\hat{b}^TAc+\hat{b}^T\hat{c}=\frac{1}{24}\)
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\(p = 5\)
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\( b^Tc^4 + 4\hat{b}^Tc^3 =\frac{1}{5}\)
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\(b^Tc^2Ac + b^Tc^2\hat{c}+\hat{b}^Tc^3+2\hat{b}^TcAc+2\hat{b}^Tc\hat{c}=\frac{1}{10}\)
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\(b^TcAc^2+2b^Tc\hat{A}c+\hat{b}^Tc^3+\hat{b}^T Ac^2+2\hat{b}^T\hat{A}c=\frac{1}{15}\)
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\(b^TcA^2c+b^TcA\hat{c}+b^Tc\hat{A}c+\hat{b}^TcAc+\hat{b}^Tc\hat{c}+ \hat{b}^TA^2c+\hat{b}^TA\hat{c}+\hat{b}^T\hat{A}c=\frac{1}{30}\)
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\(b^T(Ac)(Ac)+2b^T\hat{c}Ac+b^T\hat{c}^2+ 2\hat{b}^TcAc+2\hat{b}^Tc\hat{c}=\frac{1}{20}\)
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\( b^TAc^3+3b^T\hat{A}c^2+\hat{b}^Tc^3=\frac{1}{20}\)
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\(b^TA(cAc)+b^TA(c\hat{c})+b^T\hat{A}c^2+b^T\hat{A}Ac+b^T\hat{A}\hat{c} + \hat{b}^TcAc+\hat{b}^Tc\hat{c}=\frac{1}{40}\)
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\(b^TA^2c^2+2b^TA\hat{A}c+b^T\hat{A}c^2+\hat{b}^T Ac^2+2\hat{b}^T\hat{A}c=\frac{1}{60}\)
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\(b^TA^3 c+b^TA^2 \hat{c}+b^T A \hat{A}c+b^T\hat{A}Ac+b^T\hat{A}\hat{c} +\hat{b}^TA^2c +\hat{b}^TA\hat{c}+\hat{b}^T\hat{A}c=\frac{1}{120} \)
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