Depth averaged 2D model formulation

Governing equations

The two dimensional model solves the depth averaged shallow water equations (7)-(8). The prognostic variables are the water elevation \(\eta\) and depth averaged velocity \(\bar{\mathbf{u}}\). The corresponding fields in Thetis are called 'elev_2d' and 'uv_2d'.

Wetting and drying

Wetting and drying is included through the modified bathymetry formulation of Karna et al. (2011). The modified equations are given by (13)-(14). The \(\alpha\) parameter associated with the wetting and drying scheme can be specified by the ModelOptions2d.wetting_and_drying_alpha option. It can also be computed automatically, by setting the ModelOptions2d.use_automatic_wetting_and_drying_alpha option to True.

Spatial discretization

Thetis supports different finite element discretizations, summarised in the table below.

Element Family

Name

Degree n

\(\bar{\mathbf{u}}\) space

\(\eta\) space

Equal order DG

'dg-dg'

1, 2

P(n)DG

P(n)DG

Raviart-Thomas DG

'rt-dg'

1, 2

RT(n+1)

P(n)DG

P1DG-P2

'dg-cg'

1

P(n)DG

P(n+1)

Brezzi-Douglas-Marini DG

'bdm-dg'

1, 2

BDM(n+1)

P(n)DG

Table 1. Finite element families for polynomial degree n.

The element family and polynomial degree are set by the ModelOptions2d.element_family and ModelOptions2d.polynomial_degree options.

Lax-Friedrichs stabilization is used by default and may be controlled using the ModelOptions2d.use_lax_friedrichs_velocity option. The scaling parameter used by this scheme may be controlled using the ModelOptions2d.lax_friedrichs_velocity_scaling_factor option.

Temporal discretization

Thetis supports different time integration methods, set by the ModelOptions2d.swe_timestepper_type option.

Time integrator

Thetis class

Unconditionally stable

Description

'ForwardEuler'

ForwardEuler

No

Forward Euler method

'BackwardEuler'

BackwardEuler

Yes

Backward Euler method

'CrankNicolson'

CrankNicolson

Yes

Crank-Nicolson method

'DIRK22'

DIRK22

Yes

DIRK(2,3,2) method

'DIRK33'

DIRK33

Yes

DIRK(3,4,3) method

'SSPRK33'

SSPRK33

No

SSPRK(3,3) method

'SSPIMEX'

IMEXLPUM2

No

LPUM2 SSP IMEX scheme

'PressureProjectionPicard'

PressureProjectionPicard

No

Efficient pressure projection solver

'SteadyState'

SteadyState

–

Solves equations in steady state

Table 2. Time integration methods for 2D model.

Model time step is defined by the ModelOptions2d.timestep option.

For explicit solvers, Thetis can also estimate the maximum stable time step based on the mesh resolution, used element family and time integration scheme. To use this feature, the user should provide the maximal horizontal velocity scale with ModelOptions2d.horizontal_velocity_scale option and set ModelOptions2d.timestepper_options.use_automatic_timestep to True.