Techniques for the numerical solution of partial differential equations

2 Techniques for the numerical solution of partial differential equations 

An alternative technique is the method of orthogonal collocation. The method involves several number of spatial collocation points at which the partial derivatives are formulated in terms of ODEs. In other words, the method allows the discretization of PDEs into an ODE set over the collocation points, with the number of ODEs being determined by the number points used. The resulting ODE set can then be solved by one of the methods outlined in Section 11.1. Using higher number of collocation points improves the spatial discretization but increases the cost of ODE solution, which is less of an issue with the help of fast 

The solver, second element of a CFD package, transforms the equations of transport phenomena into a system of algebraic equations via discretization and solves them by an iterative approach. These steps can be carried out either by the FEM or by the FVM that are explained in the following sections. The following balance equation in an arbitrarily chosen element or cell can be written for a general flow variable (p, for example, a velocity component or enthalpy [12]  

Outcomes of the solvers are analyzed by using the postprocessors, the collection of tools that, in general, allows detailed visualization of the computational domain and the grid architecture, editing and refining the 2D/three-dunensional (3D) plots of the distributions of dependent variables (e.g., velocity, temperature, species concentration) along the computational domain and displaying time-variable results in animated forms. 

---