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 ut. 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:

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

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

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 NPDE 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.