Like STOMP, eSTOMP uses Newton-Raphson linearization to solve the set of nonlinear algebraic equations formed by the discretized governing equations. This scheme has two major computational components: (1) to compute the Jacobian matrix and solution vector elements and (2) to solve the resulting linear system of equations.
The nonlinearities in the coupled flow and energy transport system of equations are resolved through the application of the iterative Newton-Raphson technique. The Newton-Raphson linearization technique is an iterative method for solving nonlinear algebraic equations of the form
where f(x) is differentiable in x. The linearization concept approximates f(x) with suitable tangents.
NR Iteration (Conceptual Plot)
Each iteration yields a new estimate of x as the intersection of the tangent to the function f(x) at the previous estimate of x and the abscissa axis, according to
In this formulation f(x) is considered the equation residual. For convergent systems, the residual decreases quadratically with iteration. In multiple variable systems, as with the coupled flow and energy transport system, the scalar function, f(x), is replaced with a vector function R(x), according to
The vector function, R(x), represents the system of nonlinear algebraic equations produced from discretizing the conservation equations for component mass and energy. The vector of unknowns, x, represents the set of primary variables for the system, which are determined by the operational mode and phase conditions. This can be rewritten in terms of increments to the primary variables, according to
The partial derivatives shown above form the Jacobian matrix.
For simplification, a one-dimensional system involving the solution of the water mass, air mass, and energy conservation equations will be considered. The system of linear equations that result from applying the Newton-Raphson linearization technique to this system of nonlinear algebraic equations for a computational domain with “n” nodes is shown below.
Here, each Jacobian matrix element represents a block matrix of order three, according to
Each unknown vector element represents a vector of increments to the primary variables of order three, according to
Each solution vector element represents a vector of equation residuals of order three, according to
The linear system of equations produced during Newton-Raphson iteration is solved using an iterative linear system solver. PETSc (The Portable, Extensible Toolkit for Scientific Computation) is the only option available for the linear solve for eSTOMP. PETSc is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. Like SPLIB, PETSc offers several options for sparse linear system solvers. The user is referred to the PETSc manual or website for further information on each option. eSTOMP is currently compatible with version 3.1-p8. The PETSc library can be downloaded from the PETSc website.
PETSc uses the MPI standard for all message-passing communication which enables parallel execution on multiple processors. When eSTOMP is compiled, PETSc must already be compiled so that the eSTOMP executable can link with the solver library.