swe package¶
The pyro swe hydrodynamics solver. This implements the second-order (piecewise-linear), unsplit method of Colella 1990.
Subpackages¶
Submodules¶
swe.derives module¶
swe.interface module¶
-
swe.interface.
consFlux
[source]¶ Calculate the conserved flux for the shallow water equations. In the x-direction, this is given by:
/ hu \ F = | hu^2 + gh^2/2 | \ huv /
Parameters: - idir : int
Are we predicting to the edges in the x-direction (1) or y-direction (2)?
- g : float
Graviational acceleration
- ih, ixmom, iymom, ihX : int
The indices of the height, x-momentum, y-momentum, height*species fraction in the conserved state vector.
- nspec : int
The number of species
- U_state : ndarray
Conserved state vector.
Returns: - out : ndarray
Conserved flux
-
swe.interface.
riemann_hllc
[source]¶ this is the HLLC Riemann solver. The implementation follows directly out of Toro’s book. Note: this does not handle the transonic rarefaction.
Parameters: - idir : int
Are we predicting to the edges in the x-direction (1) or y-direction (2)?
- ng : int
The number of ghost cells
- ih, ixmom, iymom, ihX : int
The indices of the height, x-momentum, y-momentum and height*species fractions in the conserved state vector.
- nspec : int
The number of species
- lower_solid, upper_solid : int
Are we at lower or upper solid boundaries?
- g : float
Gravitational acceleration
- U_l, U_r : ndarray
Conserved state on the left and right cell edges.
Returns: - out : ndarray
Conserved flux
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swe.interface.
riemann_roe
[source]¶ This is the Roe Riemann solver with entropy fix. The implementation follows Toro’s SWE book and the clawpack 2d SWE Roe solver.
Parameters: - idir : int
Are we predicting to the edges in the x-direction (1) or y-direction (2)?
- ng : int
The number of ghost cells
- ih, ixmom, iymom, ihX : int
The indices of the height, x-momentum, y-momentum and height*species fractions in the conserved state vector.
- nspec : int
The number of species
- lower_solid, upper_solid : int
Are we at lower or upper solid boundaries?
- g : float
Gravitational acceleration
- U_l, U_r : ndarray
Conserved state on the left and right cell edges.
Returns: - out : ndarray
Conserved flux
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swe.interface.
states
[source]¶ predict the cell-centered state to the edges in one-dimension using the reconstructed, limited slopes.
We follow the convection here that
V_l[i]
is the left state at the i-1/2 interface andV_l[i+1]
is the left state at the i+1/2 interface.We need the left and right eigenvectors and the eigenvalues for the system projected along the x-direction
Taking our state vector as \(Q = (\rho, u, v, p)^T\), the eigenvalues are \(u - c\), \(u\), \(u + c\).
We look at the equations of hydrodynamics in a split fashion – i.e., we only consider one dimension at a time.
Considering advection in the x-direction, the Jacobian matrix for the primitive variable formulation of the Euler equations projected in the x-direction is:
/ u 0 0 \ | g u 0 | A = \ 0 0 u /
The right eigenvectors are:
/ h \ / 0 \ / h \ r1 = | -c | r2 = | 0 | r3 = | c | \ 0 / \ 1 / \ 0 /
The left eigenvectors are:
l1 = ( 1/(2h), -h/(2hc), 0 ) l2 = ( 0, 0, 1 ) l3 = ( -1/(2h), -h/(2hc), 0 )
The fluxes are going to be defined on the left edge of the computational zones:
| | | | | | | | -+------+------+------+------+------+------+-- | i-1 | i | i+1 | ^ ^ ^ q_l,i q_r,i q_l,i+1
q_r,i
andq_l,i+1
are computed using the information in zone i,j.Parameters: - idir : int
Are we predicting to the edges in the x-direction (1) or y-direction (2)?
- ng : int
The number of ghost cells
- dx : float
The cell spacing
- dt : float
The timestep
- ih, iu, iv, ix : int
Indices of the height, x-velocity, y-velocity and species in the state vector
- nspec : int
The number of species
- g : float
Gravitational acceleration
- qv : ndarray
The primitive state vector
- dqv : ndarray
Spatial derivitive of the state vector
Returns: - out : ndarray, ndarray
State vector predicted to the left and right edges
swe.simulation module¶
-
class
swe.simulation.
Simulation
(solver_name, problem_name, rp, timers=None, data_class=<class 'mesh.patch.CellCenterData2d'>)[source]¶ Bases:
simulation_null.NullSimulation
The main simulation class for the corner transport upwind swe hydrodynamics solver
-
class
swe.simulation.
Variables
(myd)[source]¶ Bases:
object
a container class for easy access to the different swe variables by an integer key
swe.unsplit_fluxes module¶
Implementation of the Colella 2nd order unsplit Godunov scheme. This is a 2-dimensional implementation only. We assume that the grid is uniform, but it is relatively straightforward to relax this assumption.
There are several different options for this solver (they are all discussed in the Colella paper).
- limiter: 0 = no limiting; 1 = 2nd order MC limiter; 2 = 4th order MC limiter
- riemann: HLLC or Roe-fix
- use_flattening: set to 1 to use the multidimensional flattening at shocks
- delta, z0, z1: flattening parameters (we use Colella 1990 defaults)
The grid indices look like:
j+3/2--+---------+---------+---------+
| | | |
j+1 _| | | |
| | | |
| | | |
j+1/2--+---------XXXXXXXXXXX---------+
| X X |
j _| X X |
| X X |
| X X |
j-1/2--+---------XXXXXXXXXXX---------+
| | | |
j-1 _| | | |
| | | |
| | | |
j-3/2--+---------+---------+---------+
| | | | | | |
i-1 i i+1
i-3/2 i-1/2 i+1/2 i+3/2
We wish to solve
we want U_{i+1/2}^{n+1/2} – the interface values that are input to the Riemann problem through the faces for each zone.
Taylor expanding yields:
n+1/2 dU dU
U = U + 0.5 dx -- + 0.5 dt --
i+1/2,j,L i,j dx dt
dU dF^x dF^y
= U + 0.5 dx -- - 0.5 dt ( ---- + ---- - H )
i,j dx dx dy
dU dF^x dF^y
= U + 0.5 ( dx -- - dt ---- ) - 0.5 dt ---- + 0.5 dt H
i,j dx dx dy
dt dU dF^y
= U + 0.5 dx ( 1 - -- A^x ) -- - 0.5 dt ---- + 0.5 dt H
i,j dx dx dy
dt _ dF^y
= U + 0.5 ( 1 - -- A^x ) DU - 0.5 dt ---- + 0.5 dt H
i,j dx dy
+----------+-----------+ +----+----+ +---+---+
| | |
this is the monotonized this is the source term
central difference term transverse
flux term
There are two components, the central difference in the normal to the interface, and the transverse flux difference. This is done for the left and right sides of all 4 interfaces in a zone, which are then used as input to the Riemann problem, yielding the 1/2 time interface values:
n+1/2
U
i+1/2,j
Then, the zone average values are updated in the usual finite-volume way:
n+1 n dt x n+1/2 x n+1/2
U = U + -- { F (U ) - F (U ) }
i,j i,j dx i-1/2,j i+1/2,j
dt y n+1/2 y n+1/2
+ -- { F (U ) - F (U ) }
dy i,j-1/2 i,j+1/2
Updating U_{i,j}:
- We want to find the state to the left and right (or top and bottom) of each interface, ex. U_{i+1/2,j,[lr]}^{n+1/2}, and use them to solve a Riemann problem across each of the four interfaces.
- U_{i+1/2,j,[lr]}^{n+1/2} is comprised of two parts, the computation
of the monotonized central differences in the normal direction
(eqs. 2.8, 2.10) and the computation of the transverse derivatives,
which requires the solution of a Riemann problem in the transverse
direction (eqs. 2.9, 2.14).
- the monotonized central difference part is computed using the primitive variables.
- We compute the central difference part in both directions before doing the transverse flux differencing, since for the high-order transverse flux implementation, we use these as the input to the transverse Riemann problem.
-
swe.unsplit_fluxes.
unsplit_fluxes
(my_data, my_aux, rp, ivars, solid, tc, dt)[source]¶ unsplitFluxes returns the fluxes through the x and y interfaces by doing an unsplit reconstruction of the interface values and then solving the Riemann problem through all the interfaces at once
The runtime parameter g is assumed to be the gravitational acceleration in the y-direction
Parameters: - my_data : CellCenterData2d object
The data object containing the grid and advective scalar that we are advecting.
- rp : RuntimeParameters object
The runtime parameters for the simulation
- vars : Variables object
The Variables object that tells us which indices refer to which variables
- tc : TimerCollection object
The timers we are using to profile
- dt : float
The timestep we are advancing through.
Returns: - out : ndarray, ndarray
The fluxes on the x- and y-interfaces