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