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Lec19 PDF

This document discusses finite difference methods for approximating the solution to diffusion equations. It introduces the forward time central space (FTCS) and backward time central space (BTCS) methods. The Crank-Nicolson method is also presented, which is an implicit method that is unconditionally stable. The document outlines the concepts of consistency, stability, and convergence for finite difference equations approximating partial differential equations and highlights that the time step and mesh width cannot be chosen arbitrarily for stability.

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Milind Jain
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123 views27 pages

Lec19 PDF

This document discusses finite difference methods for approximating the solution to diffusion equations. It introduces the forward time central space (FTCS) and backward time central space (BTCS) methods. The Crank-Nicolson method is also presented, which is an implicit method that is unconditionally stable. The document outlines the concepts of consistency, stability, and convergence for finite difference equations approximating partial differential equations and highlights that the time step and mesh width cannot be chosen arbitrarily for stability.

Uploaded by

Milind Jain
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Finite difference approximation of diffusion

equation
Diffusion equation: in one space dimension
Forward time central space (FTCS)
Backward time central space (BTCS)
The matrix on the L.H.S. of above equation is invertible as it is strictly
diagonally dominant.
Crank-Nicolson method
A finite difference equation is consistent with a PDE if the
difference between the PDE and FDE (ie., the T. E.) vanishes as
the sizes of the grid spacing go to zero independently.

When applied to a PDE that has a bounded solution, a FDE is


stable if it produces a bounded solution and is unstable if it
produces an unbounded one.

A fd method is convergent if the solution of the FDE


approaches the exact solution of the PDE as the sizes of the
grid spacing approach zero.
Consider the equation u_t=u_xx
Apparently, we cannot choose the time step/mesh width arbitrary!
So the instability must be due to choices in the numerical scheme.
j-1
Home work:

(1) Derive the truncation error of both BTCS and Crank-Nicolson


scheme and find the order of accuracy of the scheme.
Relative advantages and disadvantages of the two
approaches

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