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

Abstract

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 (p2) 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.