Spectral methods are effective methods for the solution of time-dependent partial differential equations (PDEs) with periodic boundary condition. In this article, a Fourier-Galerkin approach is used leading to an approximation method with two steps, consisting of the truncation of the Fourier-Galerkin series and the solution of the resulting ordinary differential equation with a Runge-Kutta (RK) solver. Both execution steps do not only influence the numerical accuracy of the final solution but also the performance and energy behavior of the solution process. However, a high accuracy and a low energy consumption are desirable but incompatible features. In this article, the effect of influencing parameters on the accuracy and the energy consumption is investigated. Also different RK methods with different accuracy order are considered. In particular, the question which combination of the influencing parameters and chosen RK method leads to the smallest energy consumption for a required numerical accuracy is addressed. Experimental evaluations on multicore processors are presented.