Three-dimensional simulation of high compressible flows using a multi-time-step integration technique with subcycles

An algorithm to simulate three-dimensional high compressible flows using the finite element method and a multi-time-step integration technique with subcycles is presented in this work. An explicit two-step Taylor–Galerkin scheme is adopted to integrate in time the continuum equations. When explicit schemes are used, the time-steps must satisfy the CFL stability conditions. If the smallest critical time-step is adopted uniformly for the whole domain, the integration scheme may consume a large amount of CPU time. Multi-time-step integration techniques are very suitable in these cases because elements and nodes are separated into several groups and a different time-step is assigned to each group. In this way, each group of elements is integrated with a time interval which is much closer to the critical time-steps of the elements in the group. This results in great computational savings, mainly when element sizes and properties are very different, leading to significant differences in the local critical time-step values. Multi-time-steps integration techniques are also very useful in transient problems, taking into account that at the end of each subcycle, values of the unknowns at the same time level are obtained. The multi-time-step algorithm is applied to analyze the supersonic flow (Mach=8.5) past a sphere immersed in a non-viscous flow, and the results and computational performance are compared with those obtained when a uniform time-step is used over the whole domain.