Time-marching Algorithms

In this research, the Lax, MacCormack, and Four-Stage Runge-Kutta schemes were chosen as time marching algorithms. For all three explicit methods, the residual is defined in equation 5, is the cell number, is the time level, and is the time step. The Lax scheme is first-order accurate, and the time stepping scheme is:



The MacCormack is a two-step time integration scheme of second-order accurate. In this method, the predictor and corrector step has the following forms:

Predictor step:



Corrector step:



Finally, the four-stage Runge-Kutta time-stepping scheme is fourth-order accurate in time and consists of the following steps:






In the four-stage Runge-Kutta, the starred quantities denote intermediate time levels between and .