naginterfaces.library.pde.dim1_parab_coll¶
- naginterfaces.library.pde.dim1_parab_coll(m, ts, tout, pdedef, bndary, u, xbkpts, npoly, uinit, acc, comm, itask, itrace, ind, data=None, io_manager=None, spiked_sorder='C')[source]¶
dim1_parab_coll
integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable. The spatial discretization is performed using a Chebyshev collocation method, and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a backward differentiation formula method.For full information please refer to the NAG Library document for d03pd
https://www.nag.com/numeric/nl/nagdoc_29.2/flhtml/d03/d03pdf.html
- Parameters
- mint
The coordinate system used:
Indicates Cartesian coordinates.
Indicates cylindrical polar coordinates.
Indicates spherical polar coordinates.
- tsfloat
The initial value of the independent variable .
- toutfloat
The final value of to which the integration is to be carried out.
- pdedefcallable (p, q, r, ires) = pdedef(t, x, u, ux, ires, data=None)
must compute the values of the functions , and which define the system of PDEs.
The functions may depend on , , and and must be evaluated at a set of points.
- Parameters
- tfloat
The current value of the independent variable .
- xfloat, ndarray, shape
Contains a set of mesh points at which , and are to be evaluated. and contain successive user-supplied break-points and the elements of the array will satisfy .
- ufloat, ndarray, shape
contains the value of the component where , for , for .
- uxfloat, ndarray, shape
contains the value of the component where , for , for .
- iresint
Set to or .
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- Returns
- pfloat, array-like, shape
must be set to the value of where , for , for , for .
- qfloat, array-like, shape
must be set to the value of where , for , for .
- rfloat, array-like, shape
must be set to the value of where , for , for .
- iresint
Should usually remain unchanged. However, you may set to force the integration function to take certain actions as described below:
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to = 6.
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set when a physically meaningless input or output value has been generated. If you consecutively set ,
dim1_parab_coll
returns to the calling function with the error indicator set to = 4.
- bndarycallable (beta, gamma, ires) = bndary(t, u, ux, ibnd, ires, data=None)
must compute the functions and which define the boundary conditions as in equation (3).
- Parameters
- tfloat
The current value of the independent variable .
- ufloat, ndarray, shape
contains the value of the component at the boundary specified by , for .
- uxfloat, ndarray, shape
contains the value of the component at the boundary specified by , for .
- ibndint
Specifies which boundary conditions are to be evaluated.
must set up the coefficients of the left-hand boundary, .
must set up the coefficients of the right-hand boundary, .
- iresint
Set to or .
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- Returns
- betafloat, array-like, shape
must be set to the value of at the boundary specified by , for .
- gammafloat, array-like, shape
must be set to the value of at the boundary specified by , for .
- iresint
Should usually remain unchanged. However, you may set to force the integration function to take certain actions as described below:
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to = 6.
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set when a physically meaningless input or output value has been generated. If you consecutively set ,
dim1_parab_coll
returns to the calling function with the error indicator set to = 4.
- ufloat, array-like, shape
If the value of must be unchanged from the previous call.
- xbkptsfloat, array-like, shape
The values of the break-points in the space direction. must specify the left-hand boundary, , and must specify the right-hand boundary, .
- npolyint
The degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of break-points.
- uinitcallable u = uinit(npde, x, data=None)
must compute the initial values of the PDE components , for , for .
- Parameters
- npdeint
The number of PDEs in the system.
- xfloat, ndarray, shape
, contains the values of the th mesh point, for .
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- Returns
- ufloat, array-like, shape
must be set to the initial value , for , for .
- accfloat
A positive quantity for controlling the local error estimate in the time integration. If is the estimated error for at the th mesh point, the error test is:
- commdict, communication object, modified in place
Communication structure.
On initial entry: need not be set.
- itaskint
Specifies the task to be performed by the ODE integrator.
Normal computation of output values at .
One step and return.
Stop at first internal integration point at or beyond .
- itraceint
The level of trace information required from
dim1_parab_coll
and the underlying ODE solver. may take the value , , , or .No output is generated.
Only warning messages from the PDE solver are printed.
Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
If , is assumed and similarly if , is assumed.
The advisory messages are given in greater detail as increases. You are advised to set , unless you are experienced with submodule
ode
.- indint
Indicates whether this is a continuation call or a new integration.
Starts or restarts the integration in time.
Continues the integration after an earlier exit from the function. In this case, only the argument should be reset between calls to
dim1_parab_coll
.- dataarbitrary, optional
User-communication data for callback functions.
- io_managerFileObjManager, optional
Manager for I/O in this routine.
- spiked_sorderstr, optional
If is spiked (i.e., has unit extent in all but one dimension, or has size ), selects the storage order to associate with it in the NAG Engine:
- spiked_sorder =
row-major storage will be used;
- spiked_sorder =
column-major storage will be used.
Two-dimensional arrays returned from callback functions in this routine must then use the same storage order.
- Returns
- tsfloat
The value of corresponding to the solution values in . Normally .
- ufloat, ndarray, shape
will contain the computed solution at .
- xfloat, ndarray, shape
The mesh points chosen by
dim1_parab_coll
in the spatial direction. The values of will satisfy .- indint
.
- Raises
- NagValueError
- (errno )
On entry, on initial entry .
Constraint: on initial entry .
- (errno )
On entry, .
Constraint: or .
- (errno )
On entry, , , and .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, , and .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, and .
Constraint: or
- (errno )
On entry, .
Constraint: , or .
- (errno )
On entry, .
Constraint: , or .
- (errno )
On entry, is too small: and .
- (errno )
On entry, and .
Constraint: .
- (errno )
In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting in or .
- (errno )
Singular Jacobian of ODE system. Check problem formulation.
- (errno )
was too small to start integration: .
- (errno )
set to an invalid value in call to or .
- (errno )
Serious error in internal call to an auxiliary. Increase for further details.
- (errno )
Error during Jacobian formulation for ODE system. Increase for further details.
- (errno )
Flux function appears to depend on time derivatives.
- Warns
- NagAlgorithmicWarning
- (errno )
Underlying ODE solver cannot make further progress from the point with the supplied value of . , .
- (errno )
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as : .
- (errno )
In evaluating residual of ODE system, has been set in or . Integration is successful as far as : .
- (errno )
Integration completed, but a small change in is unlikely to result in a changed solution. .
- Notes
dim1_parab_coll
integrates the system of parabolic equations:where , and depend on , , , and the vector is the set of solution values
and the vector is its partial derivative with respect to . Note that , and must not depend on .
The integration in time is from to , over the space interval , where and are the leftmost and rightmost of a user-defined set of break-points . The coordinate system in space is defined by the value of ; for Cartesian coordinates, for cylindrical polar coordinates and for spherical polar coordinates.
The system is defined by the functions , and which must be specified in .
The initial values of the functions must be given at , and must be specified in .
The functions , for , which may be thought of as fluxes, are also used in the definition of the boundary conditions for each equation. The boundary conditions must have the form
where or .
The boundary conditions must be specified in . Thus, the problem is subject to the following restrictions:
, so that integration is in the forward direction;
, and the flux must not depend on any time derivatives;
the evaluation of the functions , and is done at both the break-points and internally selected points for each element in turn, that is , and are evaluated twice at each break-point. Any discontinuities in these functions must, therefore, be at one or more of the break-points ;
at least one of the functions must be nonzero so that there is a time derivative present in the problem;
if and , which is the left boundary point, then it must be ensured that the PDE solution is bounded at this point. This can be done by either specifying the solution at or by specifying a zero flux there, that is and . See also Further Comments.
The parabolic equations are approximated by a system of ODEs in time for the values of at the mesh points. This ODE system is obtained by approximating the PDE solution between each pair of break-points by a Chebyshev polynomial of degree . The interval between each pair of break-points is treated by
dim1_parab_coll
as an element, and on this element, a polynomial and its space and time derivatives are made to satisfy the system of PDEs at spatial points, which are chosen internally by the code and the break-points. In the case of just one element, the break-points are the boundaries. The user-defined break-points and the internally selected points together define the mesh. The smallest value that can take is one, in which case, the solution is approximated by piecewise linear polynomials between consecutive break-points and the method is similar to an ordinary finite element method.In total there are mesh points in the spatial direction, and ODEs in the time direction; one ODE at each break-point for each PDE component and () ODEs for each PDE component between each pair of break-points. The system is then integrated forwards in time using a backward differentiation formula method.
- References
Berzins, M, 1990, Developments in the NAG Library software for parabolic equations, Scientific Software Systems, (eds J C Mason and M G Cox), 59–72, Chapman and Hall
Berzins, M and Dew, P M, 1991, Algorithm 690: Chebyshev polynomial software for elliptic-parabolic systems of PDEs, ACM Trans. Math. Software (17), 178–206
Zaturska, N B, Drazin, P G and Banks, W H H, 1988, On the flow of a viscous fluid driven along a channel by a suction at porous walls, Fluid Dynamics Research (4)