On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

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• 作者：Truong Nguyen-Ba
• 关键词：Method for stiff ODE’s ; Hermite–Birkhoff method ; Endpoint error ; Number of steps ; Stiff DETEST problems ; Confluent vandermonde ; type systems
• 刊名：Numerical Algorithms
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：71
• 期：4
• 页码：855-888
• 全文大小：579 KB
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• 作者单位：Truong Nguyen-Ba (1)
  
  1. Department of Mathematics and Statistics, University of Ottawa, K1N 6N5, Ottawa, Ontario, Canada
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

Variable-step (VS) 4-stage k-step Hermite–Birkhoff (HB) methods of order p = (k + 2), p = 9, 10, denoted by HB (p), are constructed as a combination of linear k-step methods of order (p − 2) and a diagonally implicit one-step 4-stage Runge–Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge–Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop L(a)-stable methods of order up to 11 with a > 63°. Fast algorithms are developed for solving these systems in O (p 2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsizes of these methods are controlled by a local error estimator. HB(p) of order p = 9 and 10 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8) and Ebadi et al. hybrid backward differentiation formulae of order 10 and 12, HBDF(10-12) in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.