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

• Published in: Numerical algorithms 2016-04, Vol.71 (4), p.855-888
• Main Author: Nguyen-Ba, Truong
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Basis functions; Computer Science; Differential equations; Differentiation; Formulas (mathematics); Interpolation; Mathematical analysis; Mathematical models; Numeric Computing; Numerical Analysis; Ordinary differential equations; Original Paper; Polynomials; Problem solving; Runge-Kutta method; Solvers; Taylor series; Theory of Computation
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-015-0027-1

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.