{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec5_notes

# lec5_notes - 16.920J/SMA 5212 Numerical Methods for Partial...

This preview shows pages 1–6. Sign up to view the full content.

16.920J/SMA 5212 Numerical Methods for Partial Differential Equations Lecture 5 Finite Differences: Parabolic Problems B. C. Khoo Thanks to Franklin Tan 19 February 2003

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
16.920J/SMA 5212 Numerical Methods for PDEs 2 OUTLINE Governing Equation Stability Analysis 3 Examples Relationship between σ and λ h Implicit Time-Marching Scheme Summary Slide 2 GOVERNING EQUATION Consider the Parabolic PDE in 1-D If υ viscosity Diffusion Equation If υ thermal conductivity Heat Conduction Equation Slide 3 STABILITY ANALYSIS Discretization Keeping time continuous, we carry out a spatial discretization of the RHS of [ ] 2 2 0, u u x t x υ π = 0 subject to at 0, at u u x u u x π π = = = = 0 x = x π = 0 u u π ( 29 , ? u x t = 2 2 u u t x υ = 0 x = x π = 0 x 1 x 2 x 1 N x - N x
16.920J/SMA 5212 Numerical Methods for PDEs 3 Slide 4 STABILITY ANALYSIS Discretization which is second-order accurate. Schemes of other orders of accuracy may be constructed. Slide 5 Construction of Spatial Difference Scheme of Any Order p The idea of constructing a spatial difference operator is to represent the spatial differential operator at a location by the neighboring nodal points, each with its own weightage. The order of accuracy, p of a spatial difference scheme is represented as ( 29 p O x . Generally, to represent the spatial operator to a higher order of accuracy, more nodal points must be used. Consider the following procedure of determining the spatial operator j du dx up to the order of accuracy ( 29 2 O x : There is a total of 1 grid points such that , 0,1,2, .... , j N x j x j N + = = 2 2 Use the Central Difference Scheme for u x 2 1 1 2 2 2 2 ( ) j j j j u u u u O x x x + - - + = + j - 2 j - 1 j j +1 j+2 j du dx

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
16.920J/SMA 5212 Numerical Methods for PDEs 4 1. Let j du dx be represented by u at the nodes j - 1, j , and j +1 with 1 α - , 0 α and 1 α being the coefficients to be determined, i.e. ( 29 1 1 0 1 1 p j j j j du u u u O x dx α α α - - + + + + = 2. Seek Taylor Expansions for 1 j u - , j u and 1 j u + about j u and present them in a table as shown below. (Note that p is not known a priori but is determined at the end of the analysis when the α ’s are made known.) u j u j u j ′′ u j ′′′ j u 0 1 0 0 1 1 j u α - - 1 α - 1 x α - -∆ ⋅ 2 1 1 2 x α - 3 1 1 6 x α - - 0 j u α 0 α 0 0 0 1 1 j u α + 1 α 1 x α ∆ ⋅ 2 1 1 2 x α 3 1 1 6 x α 1 1 k j k j k k u u α = + =- + 1 S 2 S 3 S 4 S ( 1 ) This column consists of all the terms on the LHS of (1). Each cell in this row comprises the sum of its corresponding column.
16.920J/SMA 5212 Numerical Methods for PDEs 5 where 1 1 2 3 4 1 .... k j k j k k u u S S S S α = + =- + = + + + + 3. Make as many i S ’s as possible vanish by choosing appropriate k α ’s.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}