S.P. Venkateshan

Chapter 1 – Preliminaries

This chapter is a stand alone that deals with numerical preliminaries. Specifically this chapter talks about precision in numerical calculations made using a digital computer and its effect on numerical calculations. This is followed by a brief introduction to MATLAB programming. The rest of the book is arranged in four modules as follows: n• nModule I: System of equations and eigenvalues n n• nModule II: Interpolation, differentiation and integration n n• nModule III: Ordinary differential equations n n• nModule IV: Partial differential equations

Computation of Eigenvalues

This chapter deals with eigenvalues and eigenvectors of matrices. The material here is a sequel to Chapter 2 dealing with the solution of linear equations. Eigenvalues are very important since many engineering problems naturally lead to eigenvalue problems. When the size of a matrix is large special numerical methods are necessary for obtaining eigenvalues and eigenvectors.

Advection and Diffusion Equations

This chapter considers an important class of problems where advection diffusion come together. Numerical schemes for advection diffusion terms have different limitations. Hence it is a challenge to evolve schemes where both are present. n nStability and accuracy are dependent on the specific schemes used for spatial and temporal discretization. Hence the study of stability issues and error analysis using methods such as von Neumann stability analysis and Taylor series are important. These are discussed in sufficient detail so that the reader can look independently at such issues in new situations he/she may come across. Many worked examples bring out the features of different numerical schemes.

Boundary Value Problems (ODE)

Second higher order ordinary differential equations may also be formulated as problems defined over finite or semi-infinite domains. In such cases the function its derivatives may be specified at either ends of the interval defining the domain. The ODE becomes a boundary value problem (BVP). n nBVPs occur in applications such as heat transfer, electromagnetism, fluid mechanics and, in general, in many field problems in one space dimension. Hence the solution of such equations is important. The following numerical methods are considered in this chapter: n• nShooting method n n• nFinite difference method (FDM) n n• nMethod of weighted residuals (MWR) n n• nCollocation method n n• nFinite element and finite volume methods

Regression or Curve Fitting

Regression analysis consists of expressing data, usually experimental data, in a succinct form as a formula thus giving a global interpolating formula. Experimental data is usually error ridden and hence analysis has to identify a possible regression law or model that best fits the data. The regression formula does not attempt to pass through all data points as a piecewise interpolating function is expected to do. Invariably the regression formula does not agree with the tabulated values exactly, but only in an average sense. The regression model is expected to pass close to all the data points and yield the smallest overall error, defined in a suitable fashion.

Initial Value Problems

Ordinary differential equations ( ODE ) occur very commonly in analysis of problems of engineering interest. Analytical solution of such equations is many times difficult or the evaluation using closed form solution itself may be as laborious as a numerical solution. Hence numerical solution of ODE is a topic of much practical utility. We develop numerical methods to solve first order ODEs and extend these to solve higher order ODEs, as long as they are initial value problems ( IVP ) . The following methods of solution of IVPs will be discussed here: • Euler method • Modified Euler or Heun method or the second order Runge Kutta ( RK2 ) method • Runge Kutta methods • Predictor corrector methods • Backward difference formulae based methods ( BDF methods )

Interpolation in Two and Three Dimensions

Multidimensional interpolation is commonly encountered in numerical methods such as the Finite Element Method (FEM) the Finite Volume Method (FVM) used for solving partial differential equations. It is a general practice in numerical methods to discretize a two (three) dimensional domain into large number of small areas (volumes) known as elements in FEM volumes in FVM. These methods assume a functional form for variable within each sub domain based on the nodal values solve for the variables at the nodes. Development of multidimensional interpolation has also applications in computer graphics where surfaces are represented by Bezier curves NURBS (Non-uniform rational B-splines). The present chapter intends to introduce the readers to multi-dimensional interpolation, with simple examples, to be made use of in later chapters.

Solution of Linear Equations

Linear equations are encountered in all branches of engineering hence have received much attention. Geometric applications include the determination of the equation of a plane that passes through three non-collinear points, determination of the point of intersection of three non-parallel planes. Engineering applications are to be found in diverse areas such as analysis of electrical networks, conduction of heat in solids, solution of partial differential equations by finite difference finite element methods. When the number of equations are small solution may be obtained by elementary methods. For example, two or three equations may be solved easily by the use of Cramer’s rule. aWhen the number of equations become larger Cramer’s rule is cumbersome one of several alternate methods may be used. The present chapter considers some elementary cases amenable to traditional methods followed by more complex applications that require advanced techniques.

Laplace and Poisson Equations

Laplace Poisson equations occur as field problems in many areas of engineering hence have received much attention. Relaxation methods awere developed originally in order to solve such field problems. In the linear case the governing equations (Laplace or Poisson) are transformed to a set of linear equations for the nodal values are solved by the various techniques given in Chapter 2. However when nonlinearities are involved such as when the properties of the medium depends on the dependent variable, discretized equations are linearized then solved iteratively. Even though analytical solutions are sometimes possible numerical techniques have become the norm in solving elliptic PDEs.

Chapter 8 – Numerical Differentiation

In many engineering applications such as structural mechanics, heat transfer and fluid dynamics numerical methods are employed to solve the governing partial differential equations. The numerical data is available at discrete points in the computational domain. It is then necessary to use numerical differentiation to evaluate or estimate derivatives. Usually finite difference approximations are used to evaluate derivatives. Forward, backward and divided differences dealt with in section 5.3are in fact related to approximations of derivatives of a given function. We look at numerical differentiation in greater detail in this chapter.