The section describes an algorithm and a corresponding integrator function imsl_f_pde_1d_mg for solving a system of partial differential equations
Equation 1
This software is a one-dimensional
differential equation solver. It requires the user to provide initial and
boundary conditions in addition to a function for the evaluation of
. The integration method is
noteworthy due to the maintenance of grid lines in the space variable, 
. Details for choosing new grid
lines are given in Blom and Zegeling, (1994). The class of problems solved
with imsl_f_pde_1d_mg
is expressed by Equation 1 and given in more
detail by:
Equation 2
The vector 
is the solution. The
integer value 
is the number of differential
equations. The functions 
and 
can be regarded, in special cases, as
flux and source terms. The functions 
are expected to be continuous.
Allowed values for the integer 
are any of
. These are respectively for problems
in Cartesian, cylindrical or polar, and spherical coordinates. In the two
cases with
, the interval 
must not contain 
as an interior point.
The boundary conditions have the master equation form
Equation 3
In the boundary conditions the functions
and 
are continuous. In the two cases
with , with an endpoint of at 0, the finite value of the
solution at must be ensured. This
requires the specification of the solution at, or it implies that 
or 
. The initial values
satisfy
, where
is a piece-wise continuous
vector function of with 
components.
The user must pose the problem so that mathematical definitions are known for the functions
These functions are provided to the function imsl_f_pde_1d_mg in the form of two
user-supplied functions. This form of the usage interface is explained
below and illustrated with several examples. can be supplied as the input
argument u
or by an optional user-supplied function. Users comfortable with the description
of this algorithm may skip directly to the Examples section.
Description Summary
Equation 1 is approximated at 
time-dependent grid values
. Using the total differential
transforms the differential
equation to the form
Using central divided differences for the factor 
leads to the system of ordinary
differential equations in implicit form
The terms 
respectively represent
the approximate solution to the partial differential equation and the value of
at the point
. The truncation error from this
approximation is second-order in the space variable. The above ordinary differential
equations are underdetermined, so additional equations are added for determining
the time-dependent grid points. These additional equations contain
parameters that can be adjusted by the user. Often it will be necessary to
modify these parameters to solve a difficult problem. For this purpose the
following quantities are needed:
The values
are the so-called point
concentration of the grid. The parameter
denotes a spatial smoothing
value. Now the grid points are defined implicitly so that
The parameter 
denotes a time-smoothing value.
If the value
is chosen to be large, this results in
a fixed spatial grid. Increasing from its default value avoids the
error condition where grid lines cross. The divisors are defined by
The value 
determines the level of
clustering or spatial smoothing of the grid points. Decreasing from its default values also decreases
the amount of spatial smoothing. The parameters 
approximate arc length and help
determine the shape of the grid or 
distribution. The parameterprevents the grid movement from
adjusting immediately to new values of the , thereby avoiding oscillations in the
grid that cause large relative errors in the solution. This is important
when applied to solutions with steep gradients.
The discrete form of the differential equation and the smoothing equations are combined to yield the implicit system of differential equations
This is usually a stiff differential-algebraic system. It is solved using the integrator imsl_f_dea_petzold_gear, documented in this chapter. If imsl_f_dea_petzold_gear is needed during the evaluations of the differential equations or boundary conditions, it must be done in a separate thread to avoid possible problems with imsl_f_pde_1d_mg’s internal use of imsl_f_dea_petzold_gear. The only options for imsl_f_dea_petzold_gear set by imsl_f_pde_1d_mg are the Maximum BDF Order, and the absolute and relative error values, documented as IMSL_MAX_BDF_ORDER, and IMSL_ATOL_RTOL_SCALARS.
