Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower order implicit time integration schemes for local and global error estimation

Abstract

Three prominent low order implicit time integration schemes are the first order implicit Euler-method, the second order trapezoidal rule and the second order Ellsiepen method. Its advantages are stability and comparatively low computational cost, however, they require the solution of a nonlinear system of equations. This paper presents a general approach for the construction of third order Runge–Kutta methods by embedding the above mentioned implicit schemes into the class of ELDIRK-methods. These will be defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge–Kutta (DIRK) methods, with the consequence, that no additional system of equations must be solved. The main results—valid also for non-linear ordinary differential equations—are as follows: Two extra function calculations are required in order to embed the implicit Euler-method and one extra function calculation is required for the trapezoidal-rule and the Ellsiepen method, in order to obtain the third order properties, respectively. Two numerical examples are concerned with a parachute with viscous damping and a two-dimensional laser beam simulation. Here, we verify the higher order convergence behaviours of the proposed new ELDIRK-methods, and its successful performances for asymptotically exact global error estimation of so-called reversed embedded RK-method are shown.

Numerical investigations of new low‐order explicit last stage diagonal implicit Runge–Kutta schemes with the finite‐element method

“…In the current article, the novel Runge–Kutta methods are applied to the finite‐element method and heat problem. The ELDIRK schemes for second‐ and third‐order methods proposed in [1] and [2] are considered as a point of departure to bring adaptivity in time to the finite‐element method. Further improvements in numerical methods are taken into consideration in future research.…”

“…The appendix summarizes all second-order and third-order Butcher arrays in a concise manner. The Butcher arrays are taken from [1] and [2].…”

“…Thus, the selection of an appropriate integration scheme in time is crucial for stability characteristics and computational effectiveness. According to the numerical studies in [1] and [2], new low-order explicit last-stage diagonal implicit Runge-Kutta (ELDIRK) schemes have favorable convergence rates and computing efficiency. The ELDIRK methods apply Fehlberg's idea [3] to classical implicit Runge-Kutta schemes.…”