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

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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
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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
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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
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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.
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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.
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