In Hele-Shaw flow simulation

the directly solved variable is pressure and the velocity is only the post-treating result of pressure. Around the injection gate

the velocity may be very high along with reducing elemental size. This means that when the energy conservation equation is solved as a whole

the time step must be very short

otherwise

the error in the heat convection is unavoidable. The above problem can be overcome by using the operator-splitting method

in which the material at the current computing points needs to be tracked back to its position in the last time-step. However

this may lead to a new difficulty. If the elemental velocity is very high

the tracking needs to pass through a few elements and the reversetracking may fail. To solve the problem

a new algorithm named sub time step with variable size was suggested to deal with the thermalconvection in this paper

in which the sub time step using dichotomy was introduced that bounded the tracking path in a certain element. The new algorithm made the computation more simple and effective. The numerical examples showed that the new method had same accuracy as one using uniform small time step and high solving stability

but calculating time was dramatically reduced.