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Three-stage Hermite-Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs. (English) Zbl 1323.65083
Summary: Variable-step (VS) \(3\)-stage Hermite-Birkhoff (HB) methods \(\mathrm{HB}\) \((p=k+1)\) of order \(p=8\) and \(9\) are constructed as a combination of linear \(k\)-step methods of order \((p-2)\) and a diagonally implicit one-step \(3\)-stage Runge-Kutta method of order \(3\) (DIRK3) for solving stiff ordinary differential equations (ODEs). 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 \(A\)-stable methods of order up to \(5\) and \(A(\alpha)\)-stable methods of order up to \(10\). 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 stepsize of these methods are controlled by a local error estimator. When programmed in C++, \(\mathrm{HB}(p)\) of order \(p=8\) and \(9\) compare favorably with existing Cash modified extended backward differentiation formulae of order \(7\) and \(8\), MEBDF(\(7-8\)) (cf.[J. R. Cash, Numer. Math. 34, 235–246 (1980; Zbl 0411.65040)]), 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.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems
65L04 Numerical methods for stiff equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
65L70 Error bounds for numerical methods for ordinary differential equations
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