A high intensity four-vane continue wave (cw) RFQ for the Beijing Isotope Separation On-Line (BISOL) high intensity deuteron driver linac has been designed. This RFQ will accelerate the 25 mA cw deuteron beam the 3.0 MeV with an operating frequency of 162.5 MHz. Three beam dynamics design schemes based on different strategies have been proposed and detailed comparisons, including tolerance study and error analysis, are performed to evaluate them. The final design of RFQ can achieve the requirements with a length of 5.29 m and the transmission of 99.1%. The inter-vane voltage is ramped from 60 kV to 75 kV and the maximum Kilpatrick factor is 1.42, which is very modest and beneficial to cw operation mode. The RFQ electromagnetic design is carried out based on the full-length 3D model, including vane modulation, undercuts, π-mode stabilization rods (PSLs) and tuners. The requirements of correct work frequency and voltage ramp are fulfilled by varying vane width and adjusting undercuts. In addition, the π-mode stabilization rods (PSLs) are inserted and optimized to increase mode separation and enhance voltage stabilization. The rf power loss and effective shunt impendence are 118 kW and 210 kΩ⋅m, respectively. In addition, multi-physics analysis of the cavity, based on the electromagnetic codes CST MWS, HFSS and ANSYS, has been carried out to check the cooling system, the temperature rise and deformation of the cavity, as well as the frequency shift in the cw operation. Finally, the frequency sensitivities to cooling water temperature are studied to obtain water temperature tuning coefficients, which will be used for fine tuning of the RFQ cavity during operation.