Mathematics - iii (Differential Calculus) - Vol 2 | 20. Numerical Methods for PDEs (basic overview) by Abraham | Learn Smarter
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20. Numerical Methods for PDEs (basic overview)

Numerical methods are crucial for approximating solutions to Partial Differential Equations (PDEs) that model various phenomena in engineering and science. Key numerical methods include Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM), each offering unique advantages based on problem characteristics. The choice of method depends on factors such as the type of PDE, geometrical complexity, and conservation requirements, guiding effective simulation in real-world applications.

Sections

  • 20.1

    Classification Of Pdes (Recap)

    This section discusses the classification of Partial Differential Equations (PDEs) into three main types: elliptic, parabolic, and hyperbolic.

    Learning Practice

  • 20.2

    Common Numerical Methods For Pdes

    Numerical methods are vital for approximating solutions to complex Partial Differential Equations (PDEs) encountered in engineering and science.

    Learning Practice

  • 20.2.1

    Finite Difference Method (Fdm)

    The Finite Difference Method (FDM) is a numerical technique for approximating solutions to partial differential equations (PDEs) by discretizing the domain into a grid.

    Learning Practice

  • 20.2.2

    Finite Element Method (Fem)

    The Finite Element Method (FEM) is a numerical technique for approximating solutions to partial differential equations, particularly useful for complex geometries and boundary conditions.

    Learning Practice

  • 20.2.3

    Finite Volume Method (Fvm)

    The Finite Volume Method (FVM) is a numerical technique used to solve partial differential equations (PDEs) by integrating over control volumes, ensuring conservation laws are maintained.

    Learning Practice

  • 20.2.4

    Method Of Lines (Mol)

    The Method of Lines (MOL) is a numerical technique that simplifies the solution of partial differential equations (PDEs) by transforming them into systems of ordinary differential equations (ODEs) through spatial discretization.

    Learning Practice

  • 20.3

    Stability And Convergence

    This section explains the fundamental concepts of stability and convergence in numerical methods for solving Partial Differential Equations (PDEs).

    Learning Practice

  • 20.4

    Applications Of Numerical Pde Solvers

    This section explores various real-world applications of numerical PDE solvers in fields such as heat transfer, fluid dynamics, and stress analysis.

    Learning Practice

  • 20.5

    Comparison Of Methods

    The comparison of numerical methods for solving Partial Differential Equations (PDEs) highlights their distinct features regarding implementation, geometry handling, and conservation laws.

    Learning Practice

  • 20.6

    Summary

    Numerical methods provide essential tools for approximating solutions to complex Partial Differential Equations (PDEs) when analytical solutions are infeasible.

    Learning Practice

References

Unit_2_ch20.pdf

Class Notes

Memorization

What we have learnt

  • PDEs are essential in model...
  • Numerical methods provide a...
  • Key numerical methods inclu...

Final Test

Revision Tests