A two-component Bose-Einstein condensate (BEC) described by two coupled a three-dimension Gross-Pitaevskii (GP) equations is considered, where one equation has dipole-dipole interaction while the other one has only the usual s-wave contact interaction, in a cigar trap. The time-splitting and sine spectral method in space is proposed to discretize the time-dependent equations for computing the dynamics of dipolar BEC. The singularity in the dipole-dipole interaction brings significant difficulties both in mathematical analysis and in numerical simulations. Numerical results are given to show the efficiency of this method. 1. Introduction The achievement of the Bose-Einstein condensation of dilute gases in 1995 marked the beginning of a new era in atomic, molecular, and optical physics. That has attracted much attention both theoretically and experimentally. Most of their properties of these trapped quantum gases are governed by the interactions between particles in the condensate [1]. In the last several years, there has been an investigation for realizing a new kind of quantum gases with the dipolar interaction, acting between particles having a permanent magnetic or electric dipole moment. The experimental realization of a BEC of 52Cr atoms [2, 3] at the University of Stuttgart in 2005 gave new impetus to the theoretical and the numerical investigations on these novel dipolar quantum gases at low temperature. Recently more detailed and controlled experimental results have been obtained, illustrating the effects of phase separation in a multicomponent BEC [4–6]. In these papers, the studies of the binary condensates were limited to the case of s-wave interaction, while a great deal of attention has been drawn recently to the dipolar BEC. In this work, a numerical method for computing the dynamics of the two-component dipolar BEC is considered, where one equation has dipole-dipole interaction and the other has only the usual s-wave contact interaction. However, since the dipole-dipole interaction is of long range, anisotropic, and partially attractive and the computational cost in three dimensions high, the nontrivial task of achieving and controlling the dipolar BEC is thus particularly challenging. This paper is organized as follows. In Section 2, a numerical method for computing ground states is presented. In Section 3, numerical results are reported to verify the efficiency of this numerical method. Finally, some concluding remarks are drawn in Section 4. 2. Numerical Method for Computing the Dynamics 2.1. The Nonlocal Gross-Pitaevskii Equation The
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