Computational Methods of Partial Differential Equations

Paper Code: 
MAT 423B
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. Understand various numerical methods.
  2. Understand the finite difference schemes for the solution of partial differential equations along with analyzing them for consistency, stability and convergence.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

MAT 423B

 

Computational Methods of Partial Differential Equations

 (Theory)

 

 

 

The students will be able to –

 

CO158: Use discretization methods for solution of PDEs using finite difference schemes.

CO159: Analyze the consistency, stability and convergence of a given numerical scheme.

CO160: Apply various iterative techniques for solving system of algebraic equations.

CO161: Know the basics of finite element methods for the numerical solution of PDEs.

CO162: Construct computer programme using some mathematical software to test and implement numerical schemes studied in the course.

CO163: Create own algorithms to solve the differential equations numerically

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

 

Unit I: 
I
15.00
Elliptic equations: Finite difference method on 2D and 3D elliptic equation on non-uniform mesh, Finite difference methods for 2D and 3D Poisson’s equations of second and fourth order approximations, Iterative methods for 2D and 3D elliptic equations, Solution of large system of algebraic equations corresponding to discrete problems and iterative methods (Jacobi, Gauss-Seidel and SOR), Numerical methods extended method 2D and 3D bi-harmonic problems.
 
Unit II: 
II
15.00
Heat Equations: Compatibility, Consistency and convergence of the difference method, Numerical methods for one dimensional heat conduction equation: Schmidth scheme, Laasonen scheme,  Cranck Nicholson Scheme, Alternating direction implicit (ADI) methods  for 2D and 3D heat conduction equations,  Stability analysis (Energy method , Matrix method and Von-Neumann method).
 
Unit III: 
III
15.00
First order hyperbolic equation:  Conservation laws, Explicit and implicit methods for diffusion equations, Explicit and implicit difference scheme for first order hyperbolic equations and their stability analysis, System of equation for first order hyperbolic equation, Conservative form, Alternating direction implicit (ADI ) methods for 2D and 3D first order hyperbolic equation.
 
Unit IV: 
IV
Second order hyperbolic equations: Methods of characteristic  for evolution problem of hyperbolic type, Von-Neumann method for stability analysis, Explicit and implicit method for second order hyperbolic equation, Operator splitting methods for 2D and 3D wave equations and their stability analysis, Unconditional stability  analysis for second order hyperbolic equations. 
 
Unit V: 
V
15.00
Finite element method: Finite element method for second order elliptic BVPS, Finite element equation, Variational problems, Triangular and rectangular finite elements, Standard examples of finite elements, Mixed finite element methods.
 
Essential Readings: 
  • J.C. Strickwerda, Finite Difference Schemes and Partial Differential Equations, SIAM Publications, 2004.
  • C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Addison-Wesley, 1998.
  • M.K. Jain, S.R.K. Iyenger and R.K. Jain, Computational Methods for Partial Differential Equations, New Age Publications, 2015.
  • M.K. Jain, Numerical Solution of Differential Equations: Finite difference and Finite Element Approach, New Age Publications, 2018.
  • J.W. Thomas, Numerical Partial Differential Equation: Finite Difference Method, Springer and Verlag Berlin, 1998.
  • J.W. Thomas, Numerical Partial Differential Equations: Conservation Laws and Elliptical Equations, Springer and Verlag Berlin, 1999.
  • K.J. Bathe and E.L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall India 1987.
  • D.V. Griffiths and I.M. Smith, Numerical Methods for Engineers, Oxford University Press, 1993.
 
Academic Year: 
2021-2022
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