New low order Runge–Kutta schemes for asymptotically exact global error estimation of embedded methods without order reduction

Abstract

Embedded Runge–Kutta (RK) methods provide an attractive methodology for solution of initial value problems (IVP) by means of adaptive step size control. According to Fehlbergs suggestion, only one extra function calculation is required to estimate the local error of the embedded method. In the present paper, this methodology is applied to three prominent low order implicit RK-schemes. To this end, new embedded Butcher-tableaus are presented for the implicit Euler-method, the trapezoidal-rule and the midpoint-rule by means of corresponding higher order RK-schemes. In order to account for the global error, an asymptotically exact global error estimator is derived for the so-called reversed embedded RK-method. Moreover, a cause of order reduction is illustrated by aid of a weak discontinuous formulation. It is prevented in a new hybrid integration algorithm by means of a Hermite interpolation for the source term of the IVP, and its implementation renders a convenient “node-to-node” stepping scheme for a laser beam simulation. Two numerical examples are presented with the following main results: The higher order convergence behaviors of the proposed new RK-schemes are verified, its successful performances for asymptotically exact global error estimation of the reversed embedded RK-method are shown, the proposed methodology is illustrated for two different global error functions, and the desired effect of the hybrid algorithm to prevent order reduction is illustrated.