Photomultiplier tubes (PMTs) widely used in neutrino detectors are critical to reconstructing the direction of the neutrino accurately. Large photocathode coverage, compact design and good time properties for single-photoelectron light are essential performances to meet the requirements for the next generation detectors. Therefore, a novel digital optical module housing 31 3-inch. diameter PMTs is developed. In order to maximize the effective photocathode area and improve the time performance, a modified PMT with a larger photocathode area and 10 dynodes is optimized with the aid of the CST Particle Studio in this paper. Based on the Monte Carlo method and finite integration theory, the main characteristics of the modified PMT, such as uniformity, collection efficiency, gain and transit-time spread, are investigated. As the earlier stages of the PMT contribute the greatest weight to the total transit time spread, the transit time spread of single-photoelectron from photocathode to the first dynode (TTSCD1) is discussed mainly in this paper. The influences of the dynodes position on collection efficiency and TTSCD1 are analyzed. The voltage ratio scheme is also optimized slightly to obtain better collection efficiency and minimum TTSCD1. By tracing the trajectories of secondary electrons from the first to the second dynode stage, dynodes are optimized for improving timing performance and secondary electrons collection efficiency. Direct collection efficiency of secondary electrons from the first dynode to the second is improved from 56.38% to 61.01%. The effective photocathode diameter of the modified PMT is increased from traditional 72 mm to 77.5 mm and the effective area of photocathode is increased by 30.87% compared with the traditional one. What is more, the length of the new PMT is reduced to 103 mm so that the available space of the multi-PMT digital optical module is increased by 63.09% compared with the traditional one containing the high-voltage power supplies, front-end and readout electronics, refrigerating equipment, etc. The simulation results show that the mean collection efficiency of the modified PMT is ~96.40% with the supply voltage of 1000 V and it changes little by changing the supply voltage from 900 V to 1300 V. The mean transit time spread from photocathode to the first dynode is ~1 ns which is better than the transit time spread of the traditional model. And the gain can reach above 106 with a supply voltage of above 1100 V.