Resumen de Explicit exponential Runge�Kutta methods of high order for parabolic problems

Vu Thai Luan, Alexander Ostermann

• Exponential Runge�Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge�Kutta methods are available in the literature up to order 4 only. The aim of this paper is to construct a fifth-order method.

  For this purpose, we make use of a novel approach to derive the stiff order conditions for high-order exponential methods. This allows us to obtain the conditions for a method of order 5 in an elegant way. After stating the conditions, we first show that there does not exist an explicit exponential Runge�Kutta method of order 5 with less than or equal to 6 stages. Then, we construct a fifth-order method with 8 stages and prove its convergence for semilinear parabolic problems. Finally, a numerical example is given that illustrates our convergence bound.