HPC Adaptive Timestepping

The HPC solver, by default, uses adaptive timestepping to progress through the simulation. The timestep is adjusted so that it complies with the mathematical stability criteria of a 2D SWE explicit solution. In contrast, TUFLOW Classic, being an implicit solution, is not numerically bound by these criteria, and can progress through the simulation using a larger, fixed, timestep. Therefore, HPC typically uses a smaller timestep than Classic. The HPC timestep needs to conform to the limiting timestep as determined by three stability criteria or Control Numbers:

• Courant Number, Nu
  
• Wave Celerity Number, Nc
  
• Diffusion Number, Nd
  

Prior to advancing in time, the limiting timestep for all wet 2D cells is determined, and the minimum of the limiting timesteps for all cells is selected as the next timestep. The three stability criteria (Control Numbers) are summarised in Table 1 below.

Table 1 - Controlling Numbers
Control Number	Description	Expression	Control Number Limit
Courant Number (Nu)	This condition ensures that water entering one side of a 2D cell does not pass through the other side within one timestep. For this to be satisfied, the product of the water velocity (𝑢) and model timestep (∆𝑡) must be less than the cell size (∆𝑥).		<1.0
The Shallow Wave Celerity Number (Nc)	This numerical condition relates to the shallow water wave celerity (wave speed) and is derived from the fluid flow equations to represent long waves (i.e. wave length is substantially longer than the water depth). The product of the model timestep (∆𝑡) and the long wave speed (square root of the gravity (g) and water depth (h)) must be less than the cell size (∆𝑥), for the condition to be satisfied.		<1.0
Diffusion Number (Nd)	This numerical condition relates to the sub-grid scale eddy viscosity term which causes diffusion of momentum. To maintain stability the product of the eddy viscosity coefficient (ν_T) and the timestep (∆𝑡) divided by the square of the grid spacing (∆𝑥2) must remain below 0.3.		<0.3

The explicit solution scheme adopted for HPC requires all three conditions be satisfied under the limits specified. As seen in the above Table, each condition is relative to the timestep and cell size, as well as some hydraulic behaviour (velocity, depth, turbulence).