Commun. Comput. Phys., 9 (2011), pp. 668-687. Semi-Eulerian and High Order Gaussian Beam Methods for the Schrodinger Equation in the Semiclassical Regime Shi Jin 1*, Hao Wu 2, Xu Yang 31 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. 2 Department of Mathematical Sciences, Tsinghua University, Beijing 10084, China. 3 Program in Applied and Computational Mathematics, Princeton University, NJ 08544, USA. Received 9 October 2009; Accepted (in revised version) 16 March 2010 Available online 17 September 2010 doi:10.4208/cicp.091009.160310s Abstract A novel Eulerian Gaussian beam method was developed in [8] to compute the Schrodinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation. The difference lies in that, we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers. This yields effectively a local level set implementation, and the beam summation can be carried out on the initial physical space instead of the phase plane. As a consequence, it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method. Moreover, the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods, which is the topic of the second part of this paper. Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods. AMS subject classifications: 81Q20, 65M99 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Schrodinger equation, semi-Eulerian Gaussian beam method, high order methods. *Corresponding author. Email: jin@math.wisc.edu (S. Jin), hwu@tsinghua.edu.cn (H. Wu), xuyang@math.princeton.edu (X. Yang)