Computational Methods of Partial Differential Equations

Paper Code: 
24MAT423(B)
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: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

423(B)

 

 

Computational Methods of Partial Differential Equations

 (Theory)

 

 

 

CO160: Apply the numerical methods for solving 2D and 3D elliptic equations.

CO161: Analyze the consistency, stability, convergence of a given numerical scheme and evaluate numerical methods for heat conduction equation.

CO162: Apply various iterative techniques for solving system of first order Hyperbolic equations.

CO163: Explore the methods for the numerical solution of second order Hyperbolic equations.

CO164: Explore the finite element method for second order elliptic BVPS.

CO165:  Contribute effectively in course- specific interaction

 

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Elliptic equations:
15.00

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: 
Heat Equations:
15.00

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: 
First order hyperbolic equation:
15.00

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: 
Second order hyperbolic equations:
15.00

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: 
Finite element method:
15.00

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.

SUGGESTED READING

  • 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, Chapman and Hall/CRC, 2019.

e- RESOURCES

 

JOURNALS

 

Academic Year: 
2024-2025
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