Preconditioner Tools for Reaction-Transport Problems. Part I: Block-Diagonal Reaction-Based Factor without Grouping

This module is provided to assist in the generation and solution of preconditioner matrices for problems arising from reaction-transport systems. More specifically, the routines pjac_rbdpre, and psol_rbdpre are intended as auxiliary routines for the user-supplied routines pjac and psol called by daskr when the Krylov method is selected.

These routines are meant for a DAE system obtained from a system of reaction-transport PDEs, in which some of the PDE variables obey differential equations, and the rest obey algebraic equations [2, section 4]. It is assumed that the right-hand sides of all the equations have the form of a sum of a reaction term $R$ and a transport term $S$, that the transport term is discretized by finite differences, and that in the spatial discretization the PDE variables at each spatial point are kept together. Thus, the DAE system function, in terms of a dependent variable vector $u$, has the form:

$$G(t,u,\dot{u}) = I_d \dot{u} - R(t,u) - S(t,u)$$

where $I_d$ is the identity matrix with zeros in the positions corresponding to the algebraic components and ones in those for the differential components, $R(t,u)$ denotes the reaction terms (spatial coupling absent), and $S(t,u)$ denotes the spatial transport terms.

As shown in Brown et al. [2], two possible preconditioners for such a system are:

• $P_R = c_J I_d - \partial R/ \partial u$, based on the reaction term $R$ alone.
• $P_{SR} = (I - c_J^{-1} \partial S/\partial u) (c_J I_d - \partial R / \partial u)$, the product of two factors (in either order), one being $P_R$ and the other being based on the spatial term $S$ alone.

The routines given here can be used to compute the reaction-based factor $P_R$. More precisely, they provide an approximation $A_R$ to $P_R$. The matrix $P_R$ is block-diagonal, with each block corresponding to one spatial point. In $A_R$, we compute each block by difference quotient approximations, by way of calls to a user-supplied subroutine rblock, that evaluates the reaction terms at a single spatial point. $A_R$ has one such block for each spatial point in the mesh. For a more economical approximation, see daskr_rbgpre which uses block-grouping in $A_R$.

The routines given here are specialized to the case of a 2D problem on a rectangular mesh in the x-y plane. However, they can be easily modified for a different problem geometry. It is also assumed that the PDE variables are ordered so that the differential variables appear first, followed by the algebraic variables.

Usage

To use these routines in conjunction with daskr, the following steps are needed:

• Set info(12) = 1 to select the Krylov iterative method and info(15) = 1 to indicate that a pjac routine exists.

• Call setup_rbdpre to set mesh-related inputs.

• Include a pjac routine, as prescribed by the daskr instructions, which calls pjac_rbdpre, and does any other Jacobian-related preprocessing needed for preconditioning. The latter routine takes as argument r0 the current value of the reaction vector $R$. This can be done, for example, by taking r0 to be rpar, and loading rpar with the vector $R$ in the last call to the res routine; in that case, the calling program must declare rpar to have length at least neq. Alternatively, insert a call to rblock within the loop over mesh points inside pjac_rbdpre.

• Include a psol routine, as prescribed by the daskr instructions, which calls psol_rbdpre for the solution of systems $A_R x = b$, and does any other linear system solving required by the preconditioner.

Example

The program example_web demonstrates the use of this preconditioner.

References

1. Peter N. Brown and Alan C. Hindmarsh, Reduced Storage Matrix Methods in Stiff ODE Systems, J. Appl. Math. & Comp., 31 (1989), pp. 40-91.
2. Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold, Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems, SIAM J. Sci. Comput., 15 (1994), pp. 1467-1488.
3. Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold, Consistent Initial Condition Calculation for Differential-Algebraic Systems, SIAM Journal on Scientific Computing 19.5 (1998): 1495-1512.