The Time Explicit Solver Runge-Kutta and Adams-Bashforth methods are discussed in this section. For the nodal discontinuous Galerkin method, it is natural and most efficient to use an explicit time-stepping method. Other situations when it can be advantageous is when using only particle tracing or wave problems together with so-called mass lumping.

The third-order Adams-Bashforth multistep method (AB3) for ut = R(u) is

where un is the solution at time tn, and k is the time step.

About Auxiliary Equation-Based Nodes is tailored toward explicit time stepping. The method is quadrature free as well as matrix free. Only element local matrices are formed. A suitable stable time step can be determined automatically by specifying the variable wahw.wtc, which should be an estimate of the maximum wave speed for the equations in the interface. The Time Explicit algorithm then translates this speed to a local so-called cell time scale. For a global time marching method like Runge-Kutta or Adams-Bashforth 3, the time step is directly related to the smallest cell time scale. When there is a large difference in cell time scales, a global time marching method is not very efficient. For this reason, there is also a local time marching method, Adams-Bashforth 3 (local), which divides the cells into groups based on the cell time scale. The groups are then time marched with different time-step sizes, making this a more efficient method.