MA4154 Advanced Numerical Methods Syllabus:
MA4154 Advanced Numerical Methods Syllabus – Anna University PG Syllabus Regulation 2021
COURSE OBJECTIVES :
To study various numerical techniques to solve linear and non-linear algebraic and transcendental equations.
To compare ordinary differential equations by finite difference and collocation methods.
To establish finite difference methods to solve Parabolic and hyperbolic equations.
To establish finite difference method to solve elliptic partial differential equations.
To provide basic knowledge in finite elements method in solving partial differential equations.
UNIT I ALGEBRAIC EQUATIONS
Systems of linear equations : Gauss elimination method – Pivoting techniques – Thomas algorithm for tri diagonal system – Jacobi, Gauss Seidel, SOR iteration methods – Conditions for convergence – Systems of nonlinear equations : Fixed point iterations, Newton’s method, Eigenvalue problems : Power method and Given’s method.
UNIT II ORDINARY DIFFERENTIAL EQUATIONS
Runge – Kutta methods for system of IVPs – Numerical stability of Runge – Kutta method – Adams – Bashforth multistep method, Shooting method, BVP: Finite difference method, Collocation method and orthogonal collocation method.
UNIT III FINITE DIFFERENCE METHOD FOR TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS
Parabolic equations : Explicit and implicit finite difference methods – Weighted average approximation – Dirichlet’s and Neumann conditions – Two dimensional parabolic equations – ADI method : First order hyperbolic equations – Method of numerical integration along characteristics – Wave equation : Explicit scheme – Stability.
UNIT IV FINITE DIFFERENCE METHODS FOR ELLIPTIC EQUATIONS
Laplace and Poisson’s equations in a rectangular region : Five point finite difference schemes, Leibmann’s iterative methods, Dirichlet’s and Neumann conditions – Laplace equation in polar coordinates : Finite difference schemes – Approximation of derivatives near a curved boundary while using a square mesh.
UNIT V FINITE ELEMENT METHOD
Basics of finite element method : Weak formulation, Weighted residual method – Shape functions for linear and triangular element – Finite element method for two point boundary value problems, Laplace and Poisson equations.
COURSE OUTCOMES :
After completing this course, students should demonstrate competency in the following skills:
Solve an algebraic or transcendental equation, linear system of equations and differential equations using an appropriate numerical method.
Solving the initial boundary value problems and boundary value problems using finite difference and finite element methods.
Solving parabolic and hyperbolic partial differential equations by finite difference methods.
Compute solution of elliptic partial differential equations by finite difference methods.
Selection of appropriate numerical methods to solve various types of problems in engineering and science in consideration with the minimum number of mathematical operations involved, accuracy requirements and available computational resources.
REFERENCES :
1. Burden, R.L., and Faires, J.D., “Numerical Analysis – Theory and Applications”,9th Edition, Cengage Learning, New Delhi, 2016.
2. Gupta S.K., “Numerical Methods for Engineers”,4th Edition, New Age Publishers, 2019.
3. Jain M. K., Iyengar S. R., Kanchi M. B., Jain, “Computational Methods for Partial Differential Equations”, New Age Publishers ,1993.
4. Sastry, S.S., “Introductory Methods of Numerical Analysis”, 5th Edition, PHI Learning, 2015.
5. Saumyen Guha and Rajesh Srivastava, “Numerical methods for Engineering and Science”, Oxford Higher Education, New Delhi, 2010.
6. Smith, G. D., “Numerical Solutions of Partial Differential Equations: Finite Difference Methods”, Clarendon Press, 1985.