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Chapter 10
Numerical Modelling
10.1 Introduction
Modelling in general refers to the process of finding solutions for physical
problems by suitable methods of approximating the variables involved. It
may be of two types: Physical and theoretical modelling. Physical modelling involves laboratory testing of the models or their prototypes under
the desired physical conditions to obtain information useful in obtaining any
empirical relations, leading to a tangible solution for the problem. Theoretical modelling process consists of forming/defining a mathematical model in
a differential or algebraic form, describing the laws of the physical problem.
This mathematical model has to be solved for a solution which in most cases
is not possible analytically, and hence requiring numerical approximation.
The first step in the development of a numerical model is the approximation
to the governing équations prescribed by a mathematical model. A systematic calibration of the numerical model is required by tuning its parameters
and further, validation against existing data and analytical results provide
the confidence to the capability of the model. A prerequisite is the error
analysis of the simulated results. The final step is the actual execution of the
numerical model to obtain solutions which are analysed and interpreted in
the form of graphs, tables, or entailed qualitative forms. Figure 10.1 depicts
flow of the steps in obtaining the numerical solution to the given mathematical représentation of the problem.
10.2 Need for Numerical Models
Theoretical calculations for obtaining solutions are applicable only in a
few practical cases. In most cases, the existing analytical solutions may be
too complex to arrive at. It is quite rare that these problems be sohed
in closed form. Even when closed-form solutions do exist, their behaviour
may still be difficult to understand. In order to obtain a better insight
to the problem, solutions are generally approximated numerically using
discretization methods.
Chapter 10
Numerical Modelling
10.1 Introduction
Modelling in general refers to the process of finding solutions for physical
problems by suitable methods of approximating the variables involved. It
may be of two types: Physical and theoretical modelling. Physical modelling involves laboratory testing of the models or their prototypes under
the desired physical conditions to obtain information useful in obtaining any
empirical relations, leading to a tangible solution for the problem. Theoretical modelling process consists of forming/defining a mathematical model in
a differential or algebraic form, describing the laws of the physical problem.
This mathematical model has to be solved for a solution which in most cases
is not possible analytically, and hence requiring numerical approximation.
The first step in the development of a numerical model is the approximation
to the governing équations prescribed by a mathematical model. A systematic calibration of the numerical model is required by tuning its parameters
and further, validation against existing data and analytical results provide
the confidence to the capability of the model. A prerequisite is the error
analysis of the simulated results. The final step is the actual execution of the
numerical model to obtain solutions which are analysed and interpreted in
the form of graphs, tables, or entailed qualitative forms. Figure 10.1 depicts
flow of the steps in obtaining the numerical solution to the given mathematical représentation of the problem.
10.2 Need for Numerical Models
Theoretical calculations for obtaining solutions are applicable only in a
few practical cases. In most cases, the existing analytical solutions may be
too complex to arrive at. It is quite rare that these problems be sohed
in closed form. Even when closed-form solutions do exist, their behaviour
may still be difficult to understand. In order to obtain a better insight
to the problem, solutions are generally approximated numerically using
discretization methods.