Fully Implicit Time Stepping Can Be Efficient on Parallel Computers

Benchmarks in high performance computing often involve a single component used in the full solution of a computational problem, such as the solution of a linear system of equations. In many cases, the choice of algorithm, which can determine the components used, is also important when solving a full problem. Numerical evidence suggests that for the Taylor-Green vortex problem at a Reynolds number of 1600, a second order implicit midpoint rule method can require less computational time than the often used linearly implicit Carpenter-Kennedy method for solving the equations of incompressible fluid dynamics for moderate levels of accuracy at the beginning of the flow evolution. The primary reason is that even though the implicit midpoint rule is fully implicit, it can use a small number of iterations per time step, and thus require less computational work per time step than the Carpenter-Kennedy method. For the same number of timesteps, the Carpenter-Kennedy method is more accurate since it uses a higher order timestepping method.