*3.3*

*Time Discretization*

The DG spatial discretization reduces the problem to a system of ordinary differential

equations which we write in the concise form:

*d*

( z h ) = *L*h ( z h , **U**h , **ζ**h )

(22)

*dt*

where **z**h is the vector of unknowns over the whole domain.

We discretize this system of equations in time using a second-order Runge-Kutta scheme,

which is equivalent to the so-called modified Euler method, written in the form:

(

)

**z**(h ) = **z**(h ) + ∆*t*m Lh z(h ) , **U**(h ) , **ζ**(h )

1

*t*

*t*

*t*

*t*

(23)

(

))

(

1 (t )

**z**(h

*t *+1)

**z *** h *+ **z**(h ) + ∆*t*m Lh z(h ) , **U**(h ) , **ζ**(h )

1

1

*t*

*t*

=

2

where ∆*t*m is the morphological time step which may be different than that of the hydrodynamic

time step, ∆*t*h , and where it is to be noted that **U **and ζ are held fixed at time *t*.

Given that explicit time stepping is used, the size of the morphological time step is

limited by a Courant-Friedrichs-Levy (CFL) condition. A direct calculation of this condition

proves difficult in practice due to the highly non-linear nature of the sediment transport function,

and instead we simply take ∆*t*m = *N * ∆*t*h , where *N *is some positive integer usually in the range

of 10 to 50, i.e. the bed is updated every 10 to 50 hydrodynamic time steps. In practice, this

approach has proven to work well for a wide variety of problems and requires little additional

computational effort. It has been estimated that using this approach the additional computational

cost for running the morphodynamic model is on the order of 2-10% of the cost of running the

hydrodynamic model alone.

16