Using PyClaw’s solvers: Classic and SharpClaw

At present, PyClaw includes two types of solvers:

• Classic: the original Clawpack algorithms, in 1/2/3D

• SharpClaw: higher-order wave propagation using WENO reconstruction and Runge-Kutta integration, in 1/2D

Solver initialization takes one argument: a Riemann solver, usually from the Riemann repository. Typically, all that is needed to select a different solver is to specify it in the problem script, e.g.

>>> from clawpack import pyclaw
>>> from clawpack import riemann
>>> solver = pyclaw.ClawSolver2D(riemann.acoustics_2D)

for the 2D acoustics equations and the Classic Clawpack solver or

>>> solver = pyclaw.SharpClawSolver2D(riemann.acoustics_2D)

for the SharpClaw solver. Most of the applications distributed with PyClaw are set up to use either solver, depending on the value of the command line option solver_type, which should be set to classic or sharpclaw.

Typically, for a given grid resolution, the SharpClaw solvers are more accurate but also more computationally expensive. For typical problems involving shocks, the Classic solvers are recommended. For problems involving high-frequency waves, turbulence, or smooth solutions, the SharpClaw solvers may give more accurate solutions at less cost. This is an active area of research and you may wish to experiment with both solvers.

Future plans include incorporation of finite-difference and discontinuous Galerkin solvers.

Key differences between the Classic and SharpClaw solvers are:

• The source term routine for the Classic solver should return the integral of the source term over a step, while the source term routine for SharpClaw should return the instantaneous value of the source term. For Classic, the source term function is set using solver.step_source, while for SharpClaw it is set using solver.dq_src. The shock-bubble interaction example shows how to use each of these.

• The solvers have different options. For a list of options and possible values, see the documentation of the ClawSolver and SharpClawSolver classes.

SharpClaw Solvers

The SharpClaw solvers are a collection of solvers that contain the functionality of the Fortran code SharpClaw, developed in David Ketcheson’s thesis. The 1D SharpClaw solver contains a pure Python implementation as well as a wrapped Fortran version. The 2D solver is in progress but not available yet. The SharpClaw solvers provide an interface similar to that of the classic Clawpack solvers, but with a few different options. The superclass solvers are not meant to be used separately but are there to provide common routines for all the Clawpack solvers. Please refer to each of the inherited classes for more info about the methods and attributes they provide each class. .. The inheritance structure is:

Example:

This is a simple example of how to instantiate and evolve a solution to a later time \(\text{t_end}\) using the 1D acoustics Riemann solver.

>>> from clawpack import pyclaw
>>> solver = pyclaw.SharpClawSolver1D()           # Instantiate a default, 1d solver

>>> solver.evolve_to_time(solution,t_end)  # Evolve the solution to t_end 

pyclaw.sharpclaw

class clawpack.pyclaw.sharpclaw.solver.SharpClawSolver(riemann_solver=None, claw_package=None)

Superclass for all SharpClawND solvers.

Implements Runge-Kutta time stepping and the basic form of a semi-discrete step (the dq() function). If another method-of-lines solver is implemented in the future, it should be based on this class, which then ought to be renamed to something like “MOLSolver”.

lim_type

Limiter(s) to be used.

• 0: No limiting.

• 1: TVD reconstruction.

• 2: WENO reconstruction.

Default = 2

weno_order

Order of the WENO reconstruction. From 1st to 17th order (PyWENO)

Default = 5

time_integrator

Time integrator to be used. Currently implemented methods:

• ‘Euler’ : 1st-order Forward Euler integration

• ‘SSP33’ : 3rd-order strong stability preserving method of Shu & Osher

• ‘SSP104’ : 4th-order strong stability preserving method Ketcheson

• ‘SSPLMM32’: 2nd-order strong stability preserving 3-step linear multistep method,

  using Euler for starting values

• ‘SSPLMM43’: 3rd-order strong stability preserving 4-step linear multistep method

  using SSPRK22 for starting values

• ‘RK’Arbitrary Runge-Kutta method, specified by setting solver.a

  and solver.b to the Butcher arrays of the method.

• ‘LMM’Arbitrary linear multistep method, specified by setting the

  coefficient arrays solver.alpha and solver.beta.

Default = 'SSP104'

char_decomp

Type of WENO reconstruction. 0: conservative variables WENO reconstruction (standard). 1: Wave-slope reconstruction. 2: characteristic-wise WENO reconstruction. 3: transmission-based WENO reconstruction. Default = 0

tfluct_solver

Whether a total fluctuation solver have to be used. If True the function that calculates the total fluctuation must be provided. Default = False

tfluct

Pointer to Fortran routine to calculate total fluctuation Default = default_tfluct (None)

aux_time_dep

Whether the auxiliary array is time dependent. Default = False

kernel_language

Specifies whether to use wrapped Fortran routines (‘Fortran’) or pure Python (‘Python’). Default = 'Fortran'.

num_ghost

Number of ghost cells. Default = 3

fwave

Whether to split the flux jump (rather than the jump in Q) into waves; requires that the Riemann solver performs the splitting. Default = False

cfl_desired

Desired CFL number. Default = 2.45

cfl_max

Maximum CFL number. Default = 2.50

dq_src

Whether a source term is present. If it is present the function that computes its contribution must be provided. Default = None

call_before_step_each_stage

Whether to call the method self.before_step before each RK stage. Default = False

accept_reject_step(state)

Decide whether to accept or not the current step. For Runge-Kutta methods the step is accepted if cfl <= cfl_max. For SSPLMM32 the choice of step-size guarantees the cfl condition is satisfied for the steps the LMM is used. Hence, we need to check the cfl condition only for the starting steps.

check_3rd_ord_cond(state, step_index, dtFE)

This routine checks the additional conditions for the 3rd-order SSPLMMs. This is a posteriori check after a step is accepted. In particular, there is a condition on the step size for the starting values and a condition on the ratio of forward Euler step sizes at very step. If the conditions are violated we muct retrieve the previous solution and discard that step; otherwise the step is accepted.

dq(state)

Evaluate dq/dt * (delta t)

dqdt(state)

Evaluate dq/dt. This routine is used for implicit time stepping.

get_dt_new()

Set size of next step depending on the time integrator and whether or not the current step was accepted.

setup(solution)

Allocate RK stage arrays or previous step solutions and fortran routine work arrays.

step(solution, take_one_step, tstart, tend)

Evolve q over one time step.

Take one step with a Runge-Kutta or multistep method as specified by solver.time_integrator.

update_saved_values(state, step_index)

Updates lists of saved function evaluations, solution values, dt and dtFE for LMMs. For 3rd-order SSPLMM additional conditions are checked if self.check_lmm_cond is set to True. If these conditions are violated, the step is rejected.

Pyclaw Classic Clawpack Solvers

The pyclaw classic clawpack solvers are a collection of solvers that represent the functionality of the older versions of clawpack. It comes in two forms, a pure python version and a python wrapping of the fortran libraries. All of the solvers available provide the same basic interface and provide the same options as the old versions of clawpack. The superclass solvers are not meant to be used separately but there to provide common routines for all the Clawpack solvers. Please refer to each of the inherited classes for more info about the methods and attributes they provide each class. .. The inheritance structure is:

Example:

This is a simple example of how to instantiate and evolve a solution to a later time \(\text{t_end}\) using the linearized 1d acoustics Riemann solver

>>> from clawpack import pyclaw
>>> solver = pyclaw.ClawSolver1D()                   # Instantiate a default, 1d solver
>>> solver.limiters = pyclaw.limiters.tvd.vanleer  # Use the van Leer limiter
>>> solver.dt = 0.0001                               # Set the initial time step
>>> solver.max_steps = 500                           # Set the maximum number of time steps

>>> solver.evolve_to_time(solution,t_end)  # Evolve the solution to t_end  

pyclaw.classic.solver

class clawpack.pyclaw.classic.solver.ClawSolver(riemann_solver=None, claw_package=None)

Generic classic Clawpack solver

All Clawpack solvers inherit from this base class.

mthlim

Limiter(s) to be used. Specified either as one value or a list. If one value, the specified limiter is used for all wave families. If a list, the specified values indicate which limiter to apply to each wave family. Take a look at pyclaw.limiters.tvd for an enumeration. Default = limiters.tvd.minmod

order

Order of the solver, either 1 for first order (i.e., Godunov’s method) or 2 for second order (Lax-Wendroff-LeVeque). Default = 2

source_split

Which source splitting method to use: 1 for first order Godunov splitting and 2 for second order Strang splitting. Default = 1

fwave

Whether to split the flux jump (rather than the jump in Q) into waves; requires that the Riemann solver performs the splitting. Default = False

step_source

Handle for function that evaluates the source term. The required signature for this function is:

def step_source(solver,state,dt)

kernel_language

Specifies whether to use wrapped Fortran routines (‘Fortran’) or pure Python (‘Python’). Default = 'Fortran'.

verbosity

The level of detail of logged messages from the Fortran solver. Default = 0.

setup(solution)

Perform essential solver setup. This routine must be called before solver.step() may be called.

step(solution, take_one_step, tstart, tend)

Evolve solution one time step

The elements of the algorithm for taking one step are:

1. Pick a step size as specified by the base solver attribute get_dt()

2. A half step on the source term step_source() if Strang splitting is being used (source_split = 2)

3. A step on the homogeneous problem \(q_t + f(q)_x = 0\) is taken

4. A second half step or a full step is taken on the source term step_source() depending on whether Strang splitting was used (source_split = 2) or Godunov splitting (source_split = 1)

This routine is called from the method evolve_to_time defined in the pyclaw.solver.Solver superclass.

Input:

• solution - (Solution) solution to be evolved

Output:

• (bool) - True if full step succeeded, False otherwise

step_hyperbolic(solution)

Take one homogeneous step on the solution.

This is a dummy routine and must be overridden.

Change to Custom BC Function Signatures

To allow better access to aux array data in the boundary condition functions both the qbc and auxbc arrays are now passed to the custom boundary condition functions. The new signature is

def my_custom_BC(state, dim, t, qbc, auxbc, num_ghost):

…

and should be adopted as soon as possible. The old signature

def my_custom_BC(state, dim, t, bc_array, num_ghost):

…

can still be used but a warning will be issued and the old signature will not be supported when version 6.0 is released. This addition is available in versions > 5.2.0.