Introduction to Subroutine PDE_1D_MG
The section describes an algorithm and a corresponding integrator subroutine PDE_1D_MG for solving a system of partial differential equations
This software is a one-dimensional
solver. It requires initial and boundary conditions in addition to values
of 
. The
integration method is noteworthy due to the maintenance of grid lines in the
space variable, x. Details for
choosing new grid lines are given in Blom and Zegeling, (1994). The class
of problems solved with PDE_1D_MG
is expressed by equations:
Equation 2
The vector
is the solution. The integer value NPDE ≥1 is the number of differential equations. The functions Rj and Qj can be regarded, in special cases, as flux and source terms. The functions
are expected to be continuous. Allowed values
m=0, m-1, and m=2
are for problems in Cartesian, cylindrical or polar, and spherical coordinates. In the two cases
m > 0 , the interval
must not contain x = 0 as an interior point.
The boundary conditions have the master equation form
Equation 3
In the boundary conditions the
are continuous functions of their arguments. In the two cases m > 0 and an endpoint occurs at 0, the finite value of the solution at x = 0 must be ensured. This requires the specification of the solution at x = 0, or implies that
or
.
The initial values satisfy
,
where u0 is a
piece-wise continuous vector function of x with 
components.
The user must pose the problem so that mathematical definitions are known for the functions
.
These functions are provided to the routine PDE_1D_MG in the form of three subroutines. Optionally, this information can be provided by reverse communication. These forms of the interface are explained below and illustrated with examples. Users may turn directly to the examples if they are comfortable with the description of the algorithm.
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