Today's most popular technology of ultrasonic flow measurement is based on the transit time principle. In this work, a numerical simulation technique applicable to the analysis and optimization of transit time flowmeters is presented. -- A flowmeter represents a large simulation problem which also requires computation of acoustic fields in moving media. For this purpose, a novel boundary integral method, the Helmholtz integral-ray tracing method (HIRM), is derived and validated. For simulation of practical acoustic devices, a hybrid simulation scheme consisting of the finite element method (FEM) and HIRM is proposed. It is demonstrated that the FEM-HIRM scheme enables an accurate and efficient simulation of a complete transit time flowmeter. -- In an industrial environment, the new simulation technique can be applied to accelerate the traditional experimental-based design of ultrasonic flowmeters.