Explicit Time-Stepping

An explicit multi-stage time-stepping scheme [1] may be used to discretize the time-derivative in Eqn. 41. The solution is advanced from time t to time with an m-stage Runge-Kutta scheme, given by:

(58)

where is the stage counter for the m-stage scheme and is the multi-stage coefficient for the ith stage. The residual is computed from the intermediate solution and, for Eqn. 45, is given by:

(59)

Residual Smoothing

Residual smoothing is a mechanism for increasing the explicit time-step size by removing the high-wave number oscillations from the residuals. The residuals for cell i are filtered through a Laplacian smoothing operator:

(60)

where the subscript j refers to the neighboring cells.

This equation is implemented using a Jacobi iteration as follows:

(61)

where represents the original (unsmoothed) residuals and represents the smoothed residuals after iteration m. The default number of smoothing operations is 2 with the under-relaxation factor  = 0.5. This is typically sufficient to allow the Courant number to be doubled.