Documentation
approximations
BoussinesqApproximation(Ra, kappa=1, g=1, rho=1, alpha=1)
Bases: BaseApproximation
Expressions for the Boussinesq approximation.
Density variations are considered small and only affect the buoyancy term. Reference parameters are typically constant. Viscous dissipation is neglected (Di << 1).
Parameters:
Name | Type | Description | Default |
---|---|---|---|
Ra |
Function | Number
|
Rayleigh number |
required |
kappa |
Function | Number
|
thermal diffusivity |
1
|
g |
Function | Number
|
gravitational acceleration |
1
|
rho |
Function | Number
|
reference density |
1
|
alpha |
Function | Number
|
coefficient of thermal expansion |
1
|
Note
The thermal diffusivity, gravitational acceleration, reference density, and coefficient of thermal expansion are normally kept at 1 when non-dimensionalised.
ExtendedBoussinesqApproximation(Ra, Di, mu=1, H=None, cartesian=True, **kwargs)
Bases: BoussinesqApproximation
Expressions for the extended Boussinesq approximation.
Extends the Boussinesq approximation by including viscous dissipation and work against gravity (both scaled with Di).
Parameters:
Name | Type | Description | Default |
---|---|---|---|
Ra |
Number
|
Rayleigh number |
required |
Di |
Number
|
Dissipation number |
required |
mu |
Number
|
dynamic viscosity |
1
|
H |
Optional[Number]
|
volumetric heat production |
None
|
cartesian |
bool
|
|
True
|
Other Parameters:
Name | Type | Description |
---|---|---|
kappa |
Number
|
thermal diffusivity |
g |
Number
|
gravitational acceleration |
rho |
Number
|
reference density |
alpha |
Number
|
coefficient of thermal expansion |
Note
The thermal diffusivity, gravitational acceleration, reference density, and coefficient of thermal expansion are normally kept at 1 when non-dimensionalised.
TruncatedAnelasticLiquidApproximation(Ra, Di, Tbar=0, chi=1, cp=1, gamma0=1, cp0=1, cv0=1, **kwargs)
Bases: ExtendedBoussinesqApproximation
Truncated Anelastic Liquid Approximation
Compressible approximation. Excludes linear dependence of density on pressure (chi)
Parameters:
Name | Type | Description | Default |
---|---|---|---|
Ra |
Number
|
Rayleigh number |
required |
Di |
Number
|
Dissipation number |
required |
Tbar |
Function | Number
|
reference temperature. In the diffusion term we use Tbar + T (i.e. T is the pertubartion) - default 0 |
0
|
chi |
Function | Number
|
reference isothermal compressibility |
1
|
cp |
Function | Number
|
reference specific heat at constant pressure |
1
|
gamma0 |
Function | Number
|
Gruneisen number (in pressure-dependent buoyancy term) |
1
|
cp0 |
Function | Number
|
specific heat at constant pressure, reference for entire Mantle (in pressure-dependent buoyancy term) |
1
|
cv0 |
Function | Number
|
specific heat at constant volume, reference for entire Mantle (in pressure-dependent buoyancy term) |
1
|
Other Parameters:
Name | Type | Description |
---|---|---|
rho |
Number
|
reference density |
alpha |
Number
|
reference thermal expansion coefficient |
mu |
Number
|
viscosity used in viscous dissipation |
H |
Number
|
volumetric heat production - default 0 |
cartesian |
bool
|
|
kappa |
Number
|
diffusivity |
g |
Number
|
gravitational acceleration |
Note
The keyword arguments may be depth-dependent, but default to 1 if not supplied.
AnelasticLiquidApproximation(Ra, Di, Tbar=0, chi=1, cp=1, gamma0=1, cp0=1, cv0=1, **kwargs)
Bases: TruncatedAnelasticLiquidApproximation
Anelastic Liquid Approximation
Compressible approximation. Includes linear dependence of density on pressure (chi)
diagnostics
GeodynamicalDiagnostics(u, p, T, bottom_id, top_id, degree=4)
Typical simulation diagnostics used in geodynamical simulations.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
u |
Function
|
Firedrake function for the velocity |
required |
p |
Function
|
Firedrake function for the pressure |
required |
T |
Function
|
Firedrake function for the temperature |
required |
bottom_id |
int
|
bottom boundary identifier. |
required |
top_id |
int
|
top boundary identifier. |
required |
degree |
int
|
degree of the polynomial approximation. |
4
|
Note
All the diagnostics are returned as a float value.
Methods:
Name | Description |
---|---|
domain_volume |
The numerical domain's volume |
u_rms |
Root-mean squared velocity |
u_rms_top |
Root-mean squared velocity along the top boundary |
Nu_top |
Nusselt number at the top boundary |
Nu_bottom |
Nusselt number at the bottom boundary |
T_avg |
Average temperature in the domain |
Source code in g-adopt/gadopt/diagnostics.py
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 |
|
energy_solver
iterative_energy_solver_parameters: dict[str, Any] = {'mat_type': 'aij', 'snes_type': 'ksponly', 'ksp_type': 'gmres', 'ksp_rtol': 1e-05, 'pc_type': 'sor'}
module-attribute
Default solver parameters for iterative solvers
direct_energy_solver_parameters: dict[str, Any] = {'mat_type': 'aij', 'snes_type': 'ksponly', 'ksp_type': 'preonly', 'pc_type': 'lu', 'pc_factor_mat_solver_type': 'mumps'}
module-attribute
Default solver parameters for direct solvers
EnergySolver(T, u, approximation, delta_t, timestepper, bcs=None, solver_parameters=None, su_advection=False)
Timestepper solver for the energy equation.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
T |
Function
|
Firedrake function for the temperature |
required |
u |
Function
|
Firedrake function for the velocity |
required |
approximation |
BaseApproximation
|
G-ADOPT base approximation describing the system of equations |
required |
delta_t |
Constant
|
the simulation time step |
required |
bcs |
Optional[dict[int, dict[str, Number]]]
|
dictionary of identifier-value pairs specifying boundary conditions |
None
|
solver_parameters |
Optional[dict[str, Any]]
|
additional solver parameters provided to PETSc |
None
|
su_advection |
bool
|
whether to use of the streamline-upwind scheme |
False
|
Source code in g-adopt/gadopt/energy_solver.py
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 |
|
setup_solver()
Sets up timestepper and associated solver.
Source code in g-adopt/gadopt/energy_solver.py
134 135 136 137 138 139 140 |
|
solve()
Advances solver.
Source code in g-adopt/gadopt/energy_solver.py
142 143 144 145 146 147 |
|
stokes_integrators
iterative_stokes_solver_parameters = {'mat_type': 'matfree', 'ksp_type': 'preonly', 'pc_type': 'fieldsplit', 'pc_fieldsplit_type': 'schur', 'pc_fieldsplit_schur_type': 'full', 'fieldsplit_0': {'ksp_type': 'cg', 'ksp_rtol': 1e-05, 'ksp_max_it': 1000, 'pc_type': 'python', 'pc_python_type': 'gadopt.SPDAssembledPC', 'assembled_pc_type': 'gamg', 'assembled_mg_levels_pc_type': 'sor', 'assembled_pc_gamg_threshold': 0.01, 'assembled_pc_gamg_square_graph': 100, 'assembled_pc_gamg_coarse_eq_limit': 1000, 'assembled_pc_gamg_mis_k_minimum_degree_ordering': True}, 'fieldsplit_1': {'ksp_type': 'fgmres', 'ksp_rtol': 0.0001, 'ksp_max_it': 200, 'pc_type': 'python', 'pc_python_type': 'firedrake.MassInvPC', 'Mp_pc_type': 'ksp', 'Mp_ksp_ksp_rtol': 1e-05, 'Mp_ksp_ksp_type': 'cg', 'Mp_ksp_pc_type': 'sor'}}
module-attribute
Default solver parameters for iterative solvers
direct_stokes_solver_parameters = {'mat_type': 'aij', 'ksp_type': 'preonly', 'pc_type': 'lu', 'pc_factor_mat_solver_type': 'mumps'}
module-attribute
Default solver parameters for direct solvers
newton_stokes_solver_parameters = {'snes_type': 'newtonls', 'snes_linesearch_type': 'l2', 'snes_max_it': 100, 'snes_atol': 1e-10, 'snes_rtol': 1e-05}
module-attribute
Default solver parameters for non-linear systems
StokesSolver(z, T, approximation, bcs=None, mu=1, quad_degree=6, cartesian=True, solver_parameters=None, J=None, constant_jacobian=False, **kwargs)
Solves the Stokes system.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
z |
Function
|
Firedrake function representing the mixed Stokes system |
required |
T |
Function
|
Firedrake function representing the temperature |
required |
approximation |
BaseApproximation
|
Approximation describing the system of equations |
required |
bcs |
Optional[dict[int, dict[str, int | float]]]
|
Dictionary of identifier-value pairs specifying boundary conditions |
None
|
mu |
Function | int | float
|
Firedrake function representing dynamic viscosity |
1
|
quad_degree |
int
|
Quadrature degree. Default value is |
6
|
cartesian |
bool
|
Whether to use Cartesian coordinates |
True
|
solver_parameters |
Optional[dict[str, str | float]]
|
Dictionary of solver parameters provided to PETSc |
None
|
J |
Optional[Function]
|
Firedrake function representing the Jacobian of the system |
None
|
constant_jacobian |
bool
|
Whether the Jacobian of the system is constant |
False
|
Source code in g-adopt/gadopt/stokes_integrators.py
135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 |
|
setup_solver()
Sets up the solver.
Source code in g-adopt/gadopt/stokes_integrators.py
217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 |
|
solve()
Solves the system.
Source code in g-adopt/gadopt/stokes_integrators.py
241 242 243 244 245 |
|
create_stokes_nullspace(Z, closed=True, rotational=False, translations=None)
Create a null space for the mixed Stokes system.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
Z |
WithGeometry
|
Firedrake mixed function space associated with the Stokes system |
required |
closed |
bool
|
Whether to include a constant pressure null space |
True
|
rotational |
bool
|
Whether to include all rotational modes |
False
|
translations |
Optional[list[int]]
|
List of translations to include |
None
|
Returns:
Type | Description |
---|---|
MixedVectorSpaceBasis
|
A Firedrake mixed vector space basis incorporating the null space components |
Source code in g-adopt/gadopt/stokes_integrators.py
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 |
|
time_stepper
Timestepper implementation, mostly copied from Thetis.
TimeIntegratorBase
Bases: ABC
Defines the API for all time integrators.
advance(t, update_forcings=None)
abstractmethod
Advances equations for one time step.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
t |
float
|
Current simulation time |
required |
update_forcings |
Optional[Function]
|
Firedrake function used to update any time-dependent boundary conditions |
None
|
Source code in g-adopt/gadopt/time_stepper.py
17 18 19 20 21 22 23 24 25 26 |
|
initialize(init_solution)
abstractmethod
Initialises the time integrator.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
init_solution |
Firedrake function representing the initial solution. |
required |
Source code in g-adopt/gadopt/time_stepper.py
28 29 30 31 32 33 34 35 36 |
|
TimeIntegrator(equation, solution, fields, dt, solution_old=None, solver_parameters=None, strong_bcs=None)
Bases: TimeIntegratorBase
Time integrator object that marches a single equation.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
equation |
BaseEquation
|
G-ADOPT equation to integrate |
required |
solution |
Function
|
Firedrake function representing the equation's solution |
required |
fields |
dict[str, Function | Constant]
|
Dictionary of Firedrake fields passed to the equation |
required |
dt |
float
|
Integration time step |
required |
solution_old |
Optional[Function]
|
Firedrake function representing the equation's solution at the previous timestep |
None
|
solver_parameters |
Optional[dict[str, Any]]
|
Dictionary of solver parameters provided to PETSc |
None
|
strong_bcs |
Optional[list[DirichletBC]]
|
List of Firedrake Dirichlet boundary conditions |
None
|
Source code in g-adopt/gadopt/time_stepper.py
54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 |
|
RungeKuttaTimeIntegrator(equation, solution, fields, dt, solution_old=None, solver_parameters=None, strong_bcs=None)
Bases: TimeIntegrator
Abstract base class for all Runge-Kutta time integrators
Source code in g-adopt/gadopt/time_stepper.py
54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 |
|
get_final_solution()
abstractmethod
Evaluates the final solution
Source code in g-adopt/gadopt/time_stepper.py
89 90 91 92 |
|
solve_stage(i_stage, t, update_forcings=None)
abstractmethod
Solves a single stage of step from t to t+dt. All functions that the equation depends on must be at right state corresponding to each sub-step.
Source code in g-adopt/gadopt/time_stepper.py
94 95 96 97 98 99 100 101 |
|
advance(t, update_forcings=None)
Advances equations for one time step.
Source code in g-adopt/gadopt/time_stepper.py
103 104 105 106 107 108 109 |
|
ERKGeneric(equation, solution, fields, dt, solution_old=None, bnd_conditions=None, solver_parameters={}, strong_bcs=None)
Bases: RungeKuttaTimeIntegrator
Generic explicit Runge-Kutta time integrator.
Implements the Butcher form. All terms in the equation are treated explicitly.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
equation |
BaseEquation
|
G-ADOPT equation to solve |
required |
solution |
Function
|
Firedrake function reperesenting the equation's solution |
required |
fields |
dict[str, Function | Constant]
|
Dictionary of Firedrake fields passed to the equation |
required |
dt |
float
|
Integration time step |
required |
solution_old |
Optional[Function]
|
Firedrake function representing the equation's solution at the previous timestep |
None
|
bnd_conditions |
Optional[dict[int, dict[str, Number]]]
|
Dictionary of boundary conditions passed to the equation |
None
|
solver_parameters |
Optional[dict[str, Any]]
|
Dictionary of solver parameters provided to PETSc |
{}
|
strong_bcs |
Optional[list[DirichletBC]]
|
List of Firedrake Dirichlet boundary conditions |
None
|
Source code in g-adopt/gadopt/time_stepper.py
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 |
|
update_solver()
Create solver objects
Source code in g-adopt/gadopt/time_stepper.py
168 169 170 171 172 173 174 175 176 |
|
update_solution(i_stage)
Computes the solution of the i-th stage
Tendencies must have been evaluated first.
Source code in g-adopt/gadopt/time_stepper.py
182 183 184 185 186 187 188 189 190 |
|
solve_tendency(i_stage, t, update_forcings=None)
Evaluates the tendency of i-th stage
Source code in g-adopt/gadopt/time_stepper.py
192 193 194 195 196 197 |
|
DIRKGeneric(equation, solution, fields, dt, solution_old=None, bnd_conditions=None, solver_parameters={}, strong_bcs=None, terms_to_add='all')
Bases: RungeKuttaTimeIntegrator
Generic implementation of Diagonally Implicit Runge Kutta schemes.
All derived classes must define the Butcher tableau coefficients :attr:a
,
:attr:b
, :attr:c
.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
equation |
BaseEquation
|
G-ADOPT equation to solve |
required |
solution |
Function
|
Firedrake function reperesenting the equation's solution |
required |
fields |
dict[str, Function | Constant]
|
Dictionary of Firedrake fields passed to the equation |
required |
dt |
float
|
Integration time step |
required |
solution_old |
Optional[Function]
|
Firedrake function representing the equation's solution at the previous timestep |
None
|
bnd_conditions |
Optional[dict[int, dict[str, Number]]]
|
Dictionary of boundary conditions passed to the equation |
None
|
solver_parameters |
Optional[dict[str, Any]]
|
Dictionary of solver parameters provided to PETSc |
{}
|
strong_bcs |
Optional[list[DirichletBC]]
|
List of Firedrake Dirichlet boundary conditions |
None
|
terms_to_add |
Optional[str | list[str]]
|
Defines which terms of the equation are to be added to this solver. Default 'all' implies ['implicit', 'explicit', 'source']. |
'all'
|
Source code in g-adopt/gadopt/time_stepper.py
231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 |
|
update_solver()
Create solver objects
Source code in g-adopt/gadopt/time_stepper.py
293 294 295 296 297 298 299 300 301 302 |
|
update_solution(i_stage)
Updates solution to i_stage sub-stage.
Tendencies must have been evaluated first.
Source code in g-adopt/gadopt/time_stepper.py
308 309 310 311 312 313 314 |
|
solve_tendency(i_stage, t, update_forcings=None)
Evaluates the tendency of i-th stage
Source code in g-adopt/gadopt/time_stepper.py
316 317 318 319 320 321 322 323 324 325 326 327 |
|
AbstractRKScheme()
Bases: ABC
Abstract class for defining Runge-Kutta schemes.
Derived classes must define the Butcher tableau (arrays :attr:a
, :attr:b
,
:attr:c
) and the CFL number (:attr:cfl_coeff
).
Currently only explicit or diagonally implicit schemes are supported.
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
a = np.array(self.a)
instance-attribute
Runge-Kutta matrix :math:a_{i,j}
of the Butcher tableau
b = np.array(self.b)
instance-attribute
weights :math:b_{i}
of the Butcher tableau
c = np.array(self.c)
instance-attribute
nodes :math:c_{i}
of the Butcher tableau
cfl_coeff()
CFL number of the scheme
Value 1.0 corresponds to Forward Euler time step.
Source code in g-adopt/gadopt/time_stepper.py
364 365 366 367 368 369 370 371 |
|
ForwardEulerAbstract()
Bases: AbstractRKScheme
Forward Euler method
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
ERKLSPUM2Abstract()
Bases: AbstractRKScheme
ERKLSPUM2, 3-stage, 2nd order Explicit Runge Kutta method
From IMEX RK scheme (17) in Higureras et al. (2014).
Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
ERKLPUM2Abstract()
Bases: AbstractRKScheme
ERKLPUM2, 3-stage, 2nd order Explicit Runge Kutta method
From IMEX RK scheme (20) in Higureras et al. (2014).
Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
SSPRK33Abstract()
Bases: AbstractRKScheme
3rd order Strong Stability Preserving Runge-Kutta scheme, SSP(3,3).
This scheme has Butcher tableau
.. math:: \begin{array}{c|ccc} 0 & \ 1 & 1 \ 1/2 & 1/4 & 1/4 & \ \hline & 1/6 & 1/6 & 2/3 \end{array}
CFL coefficient is 1.0
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
BackwardEulerAbstract()
Bases: AbstractRKScheme
Backward Euler method
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
ImplicitMidpointAbstract()
Bases: AbstractRKScheme
Implicit midpoint method, second order.
This method has the Butcher tableau
.. math:: \begin{array}{c|c} 0.5 & 0.5 \ \hline & 1.0 \end{array}
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
CrankNicolsonAbstract()
Bases: AbstractRKScheme
Crack-Nicolson scheme
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
DIRK22Abstract()
Bases: AbstractRKScheme
2-stage, 2nd order, L-stable Diagonally Implicit Runge Kutta method
This method has the Butcher tableau
.. math:: \begin{array}{c|cc} \gamma & \gamma & 0 \ 1 & 1-\gamma & \gamma \ \hline & 1/2 & 1/2 \end{array}
with :math:\gamma = (2 + \sqrt{2})/2
.
From DIRK(2,3,2) IMEX scheme in Ascher et al. (1997)
Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
DIRK23Abstract()
Bases: AbstractRKScheme
2-stage, 3rd order Diagonally Implicit Runge Kutta method
This method has the Butcher tableau
.. math:: \begin{array}{c|cc} \gamma & \gamma & 0 \ 1-\gamma & 1-2\gamma & \gamma \ \hline & 1/2 & 1/2 \end{array}
with :math:\gamma = (3 + \sqrt{3})/6
.
From DIRK(2,3,3) IMEX scheme in Ascher et al. (1997)
Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
DIRK33Abstract()
Bases: AbstractRKScheme
3-stage, 3rd order, L-stable Diagonally Implicit Runge Kutta method
From DIRK(3,4,3) IMEX scheme in Ascher et al. (1997)
Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
DIRK43Abstract()
Bases: AbstractRKScheme
4-stage, 3rd order, L-stable Diagonally Implicit Runge Kutta method
From DIRK(4,4,3) IMEX scheme in Ascher et al. (1997)
Ascher et al. (1997). Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151-167. http://dx.doi.org/10.1137/0732037
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
DIRKLSPUM2Abstract()
Bases: AbstractRKScheme
DIRKLSPUM2, 3-stage, 2nd order, L-stable Diagonally Implicit Runge Kutta method
From IMEX RK scheme (17) in Higureras et al. (2014).
Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|
DIRKLPUM2Abstract()
Bases: AbstractRKScheme
DIRKLPUM2, 3-stage, 2nd order, L-stable Diagonally Implicit Runge Kutta method
From IMEX RK scheme (20) in Higureras et al. (2014).
Higueras et al (2014). Optimized strong stability preserving IMEX Runge-Kutta methods. Journal of Computational and Applied Mathematics 272(2014) 116-140. http://dx.doi.org/10.1016/j.cam.2014.05.011
Source code in g-adopt/gadopt/time_stepper.py
373 374 375 376 377 378 379 380 381 382 383 384 385 386 |
|