TABLE OF CONTENTS If you go back to the first examples of numerical schemes in Section 4.1, which we hope you have programmed for the suggested test cases, you will have noticed the complexity of the world of numerical discretization, as some schemes happen to be completely unstable and hence useless, while others are acceptable under certain conditions. You have also noticed that the obtained numerical solutions can contain large errors, which are not always acceptable neither. This diversity raises some basic questions with regard to the properties of discretized equations:
What conditions do we have to impose on a numerical scheme to obtain an acceptable approximation to the differential problem?
Why do the various schemes have widely different behaviors and how can we predict their stability limits?
For a stable calculation, such as shown on Figure 4.1.5, how can we obtain quantitative information on the accuracy of the numerical simulation?
To provide answers to these questions, it is necessary to define, more precisely, the requirements to be imposed on a numerical scheme.
These requirements are defined as consistency, stability and convergence.
They cover different aspects of the relations between the analytical and the discretized equations, between the numerical solution and the exact, analytical solution of the differential equations representing the mathematical model.
Since all fluid flow equations can be classified as elliptic, parabolic or hyperbolic, typical examples of each of them will cover the whole range of possible systems. To analyze the properties of numerical schemes and sustain...
