Runge-Kutta (RK)

The ASP model currently has options for several different Runge-Kutta (RK) schemes. Although iterative, they permit larger time steps than the Leap-Frog method, are self-starting (albeit this is a rather minor point), can have a relatively high accuracy (e.g. 4th order RK scheme: RK4), and allow the use of odd-ordered horizontal advection schemes. Such advection schemes are dissipative, and can negate the need for (numerical) horizontal diffusion which is parameterized (this means less model parameters and also a CPU savings).

The RK4 method can be expressed as

$$\begin{split} \phi^\ast &= \phi \,+\, {\frac{\Delta t }{ 2}} ... ...ht) + \,f\left(\phi^{\ast\ast\ast}\right) \right]/6 \end{split}$$ (204)

this method is highly accurate, but the obvious drawback is that it requires 4 iterations. Some economy can obviously be made by using a 3rd order RK method, at the expense of a loss in accuracy. The literature is rife with different forms of the RK3 method. In the current model, two options are available (and they give results nearly identical to those of the RK4 method). The first RK3 method can be expressed as

$$\begin{split} \phi^\ast &= \phi \,+\, {\frac{\Delta t }{ 2}} ... ...ight) + 4 \,f\left(\phi^{\ast\ast}\right) \right]/9 \end{split}$$ (205)

The second RK3 method from Wicker and Skamarock (2002) gives essentially the same results as the above but is slightly more simple in terms of computer code and involves less memory and slightly fewer computations. It is expressed as

$$\begin{split} \phi^\ast &= \phi \,+\, {\frac{\Delta t }{ 3}} ... ... \,+\, {\Delta t} \,\,f\left(\phi^{\ast\ast}\right) \end{split}$$ (206)

This is currently the default method. For more details on this scheme, see the aforementioned reference.