PDE discretization principles

The following guidelines the PDE discretization principles.

Discretization

The section Modeling provides the routines that discretize and solve the linear systems associated to the following PDE:

• Energy equation
• Generic transport equations
• Navier-Stokes equations
• Phase advection equation
• Species transport equations

In Notus the terms of a PDE are discretized independently using building blocks provided by the section Discretization of PDE. The discretization is implicit in time, except for the advection equation which is explicit, and eventually for the advection term of the other equations. The implicit discretization of a PDE leads to a linear system Ax=b to solve, where A is the matrix, b the right-hand-side and x the solution (velocity, temperature, etc.). The numerical solution of the linear system is an approximation of the solution of the PDE.

If the advection term of an equation is discretized explicitly, the right-hand-side of the linear system is modified.

Once the linear system is created, it is solved using an iterative or direct solver (see Linear system solvers). Iterative solvers and preconditioners used are the ones of the Hypre library. They are designed for 2D and 3D problems, some of them being massively parallel and efficient on thousands of processors. The direct solver linked to Notus code is MUMPS. It has the advantage to be more robust and to compute the solution up to the computer precision, but is limited to 2D problems (or very small size 3D ones).

Some specificities exists as regards the PDE-related linear systems:

1. the vectorization of fields that maps each node index (i,j,k) into a column-matrices index '(l)';
2. the structure of the matrices, which is also vectorized.