30
2. Introduction to Numerical Methods
Fig. 2.5. Example of a 2D unstructured grid
2.4.5 Finite Approximations
Following the choice of grid type, one has to select the approximations to
be used in the discretization process. In a finite difference method, approximations for the derivatives a t the grid points have to be selected. In a finite
volume method, one has to select the methods of approximating surface and
volume integrals. In a finite element method, one has to choose the shape
functions (elements) and weighting functions.
There are many possibilities to choose from; some of those most often used
are presented in this book, some are simply mentioned and many more can be
created. The choice influences the accuracy of the approximation. It also affects the difficulty of developing the solution method, coding it, debugging it,
and the speed of the code. More accurate approximations involve more nodes
and give fuller coefficient matrices. The increased memory requirement may
require using coarser grids, partially offsetting the advantage of higher accuracy. A compromise between simplicity, ease of implementation, accuracy
and computational efficiency has t o be made. The second-order methods presented in this book were selected with this compromise in mind.
2.4.6 Solution Method
Discretization yields a large system of non-linear algebraic equations. The
method of solution depends on the problem. For unsteady flows, methods
based on those used for initial value problems for ordinary differential equations (marching in time) are used. At each time step an elliptic problem has
to be solved. Steady flow problems are usually solved by pseudo-time marching or an equivalent iteration scheme. Since the equations are non-linear,
an iteration scheme is used to solve them. These methods use successive linearization of the equations and the resulting linear systems are almost always
solved by iterative techniques. The choice of solver depends on the grid type
and the number of nodes involved in each algebraic equation. Some solvers
will be presented in Chap. 5.
2. Introduction to Numerical Methods
Fig. 2.5. Example of a 2D unstructured grid
2.4.5 Finite Approximations
Following the choice of grid type, one has to select the approximations to
be used in the discretization process. In a finite difference method, approximations for the derivatives a t the grid points have to be selected. In a finite
volume method, one has to select the methods of approximating surface and
volume integrals. In a finite element method, one has to choose the shape
functions (elements) and weighting functions.
There are many possibilities to choose from; some of those most often used
are presented in this book, some are simply mentioned and many more can be
created. The choice influences the accuracy of the approximation. It also affects the difficulty of developing the solution method, coding it, debugging it,
and the speed of the code. More accurate approximations involve more nodes
and give fuller coefficient matrices. The increased memory requirement may
require using coarser grids, partially offsetting the advantage of higher accuracy. A compromise between simplicity, ease of implementation, accuracy
and computational efficiency has t o be made. The second-order methods presented in this book were selected with this compromise in mind.
2.4.6 Solution Method
Discretization yields a large system of non-linear algebraic equations. The
method of solution depends on the problem. For unsteady flows, methods
based on those used for initial value problems for ordinary differential equations (marching in time) are used. At each time step an elliptic problem has
to be solved. Steady flow problems are usually solved by pseudo-time marching or an equivalent iteration scheme. Since the equations are non-linear,
an iteration scheme is used to solve them. These methods use successive linearization of the equations and the resulting linear systems are almost always
solved by iterative techniques. The choice of solver depends on the grid type
and the number of nodes involved in each algebraic equation. Some solvers
will be presented in Chap. 5.