Order conditions for general linear methods
We describe the derivation of order conditions, without restrictions on stage order, for general linear methods for ordinary differential equations. This derivation is based on the extension of the Albrecht approach proposed in the context of Runge–Kutta and composite and linear cyclic methods. This...
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| Veröffentlicht in: | Journal of computational and applied mathematics Jg. 290; S. 44 - 64 |
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Elsevier B.V
15.12.2015
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | We describe the derivation of order conditions, without restrictions on stage order, for general linear methods for ordinary differential equations. This derivation is based on the extension of the Albrecht approach proposed in the context of Runge–Kutta and composite and linear cyclic methods. This approach was generalized by Jackiewicz and Tracogna to two-step Runge–Kutta methods, by Jackiewicz and Vermiglio to general linear methods with external stages of different orders, and by Garrappa to some classes of Runge–Kutta methods for Volterra integral equations with weakly singular kernels. This leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods, Nordsieck methods, and general linear methods with inherent Runge–Kutta stability. Exact coefficients for several low order methods with some desirable stability properties are presented for illustration. |
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| AbstractList | We describe the derivation of order conditions, without restrictions on stage order, for general linear methods for ordinary differential equations. This derivation is based on the extension of the Albrecht approach proposed in the context of Runge–Kutta and composite and linear cyclic methods. This approach was generalized by Jackiewicz and Tracogna to two-step Runge–Kutta methods, by Jackiewicz and Vermiglio to general linear methods with external stages of different orders, and by Garrappa to some classes of Runge–Kutta methods for Volterra integral equations with weakly singular kernels. This leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods, Nordsieck methods, and general linear methods with inherent Runge–Kutta stability. Exact coefficients for several low order methods with some desirable stability properties are presented for illustration. |
| Author | Welfert, Bruno Verner, James H. Jackiewicz, Zdzisław Cardone, Angelamaria |
| Author_xml | – sequence: 1 givenname: Angelamaria orcidid: 0000-0003-2399-1137 surname: Cardone fullname: Cardone, Angelamaria email: ancardone@unisa.it organization: Dipartimento di Matematica, Università di Salerno, Fisciano (Sa), 84084, Italy – sequence: 2 givenname: Zdzisław surname: Jackiewicz fullname: Jackiewicz, Zdzisław email: jackiewicz@asu.edu organization: Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA – sequence: 3 givenname: James H. surname: Verner fullname: Verner, James H. email: jimverner@shaw.ca organization: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, B.C. V5A 1S6, Canada – sequence: 4 givenname: Bruno surname: Welfert fullname: Welfert, Bruno email: welfert@asu.edu organization: Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA |
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| Keywords | 65L05 Nordsieck methods 65L20 Order conditions Two-step Runge–Kutta formulas General linear methods |
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| SubjectTerms | Derivation Differential equations General linear methods Kernels Mathematical analysis Multistage Nordsieck methods Order conditions Runge-Kutta method Stability Two-step Runge–Kutta formulas Volterra integral equations |
| Title | Order conditions for general linear methods |
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