Truong Nguyen-Ba

Strong-Stability-Preserving, K-Step, 5- to 10-Stage, Hermite-Birkhoff Time-Discretizations of Order 12

We construct optimal k-step, 5- to 10-stage, explicit, strong-stability-preserving Hermite-Birkhoff (SSP HB) methods of order 12 with nonnegative coefficients by combining linear k-step methods of order 9 with 5- to 10-stage Runge-Kutta (RK) methods of order 4. Since these methods maintain the monotonicity property, they are well suited for solving hyperbolic PDEs by the method of lines after a spatial discretization. It is seen that the 8-step 7-stage HB methods have largest effective SSP coefficient among the HB methods of order 12 on hand. On Burgers’ equations, some of the new HB methods have larger maximum effective CFL numbers than Huang’s 7-step hybrid method of order 7, thus allowing larger step size.

Strong-Stability-Preserving 7-Stage Hermite---Birkhoff Time-Discretization Methods

Optimal, 7-stage, explicit, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) methods of orders 4 to 8 with nonnegative coefficients are constructed by combining linear k-step methods with a 7-stage Runge–Kutta (RK) method of order 4. Compared to Huang’s hybrid methods of the same order, the new methods generally have larger effective SSP coefficients and larger maximum effective CFL numbers,

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

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Strong-stability-preserving, Hermite-Birkhoff time-discretization based on k step methods and 8-stage explicit Runge-Kutta methods of order 5 and 4

, on Burgers’ equation, independently of the number k of steps, especially when k is small for both methods. Based on

Variable-step variable-order 3-stage Hermite–Birkhoff–Obrechkoff DDE solver of order 4 to 14

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A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14

, some new methods of order 4 compare favorably with other methods of the same order, including RK104 of Ketcheson. The new SSP HB methods are listed in their Shu–Osher representation in Appendix.

Pryce Structural Analysis Adapted to Hermite—Birkhoff—Taylor DAE Solvers

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.

Strong‐Stability‐Preserving Hermite—Birkhoff Time‐Discretizations of Order 4 to 12

Ruuth and Spiteri have shown, in 2002, that fifth-order strong-stability-preserving (SSP) explicit Runge-Kutta (RK) methods with nonnegative coefficients do not exist. One of the purposes of the present paper is to show that the Ruuth-Spiteri barrier can be broken by adding backsteps to RK methods. New optimal, 8-stage, explicit, SSP, Hermite-Birkhoff (HB) time discretizations of order p, p=5,6,...,12, with nonnegative coefficients are constructed by combining linear k-step methods of order (p-4) with an 8-stage explicit RK method of order 5 (RK(8, 5)). These new SSP HB methods preserve the monotonicity property of the solution and prevent error growth; therefore, they are suitable for solving hyperbolic partial differential equations (PDEs) by the method of lines. Moreover, these new HB methods have larger effective SSP coefficients and larger maximum effective CFL numbers than Huangs hybrid methods and RK methods of the same order when applied to the inviscid Burgers equation. Generally, HB methods combined with RK(8, 5) have maximum stepsize 24% larger than HB combined with RK(8, 4).

One-step 5-stage Hermite-Birkhoff-Taylor ODE solver of order 12

Abstract This article presents a solver for delay differential equations (DDEs) called HBO414DDE based on a hybrid variable-step variable-order 3-stage Hermite–Birkhoff–Obrechkoff ODE solver of order 4 to 14. The current version of our method solves DDEs with state dependent, non-vanishing, small, vanishing and asymptotically vanishing delays, except neutral type and initial value DDEs. Delayed values are computed using Hermite interpolation, small delays are dealt with by extrapolation, and discontinuities are located by a bisection method. HBO414DDE was tested on several problems and results were compared with those of known solvers like SYSDEL and the recent Matlab DDE solver ddesd and statistics show that it gives, most of the time, a smaller relative error than the other solvers for the same number of function evaluations.

Contractivity-preserving explicit Hermite–Obrechkoff ODE solver of order 13

A family of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite–Birkhoff–Taylor methods of order p = 5,6,…,14, that we denote by CPHBTRK4(d,s,p), with nonnegative coefficients are constructed by casting s-stage Runge–Kutta methods of order 4 with Taylor methods of order d. The constructed CPHBTRK4 methods are implemented using efficient variable step control and are compared to other well-known methods on a variety of initial value problems. A selected method: CP 6-stages 9-derivative HBT method of order 12, denoted by CPHBTRK412, has larger region of absolute stability than Dormand–Prince DP(8,7)13M and Taylor method T(12) of order 12. It is superior to DP(8,7)13M and T(12) methods on the basis the number of steps, CPU time, and maximum global error on several problems often used to test higher-order ODE solvers. Also, we show that the contractivity preserving property of CPHBTRK412is very efficient in suppressing the effect of the propagation of discretization errors and the new method compares positively with explicit 17 stages Runge-Kutta-Nyström pair of order 12 by Sharp et al. on a long-term integration of a standard N-body problem. The selected CPHBTRK412is listed in the Appendix.