16920J/SMA5212 Numerical Methods for Pdes Lecture 5 Finite Differences, Parabolic Problems B.C. Khoo Thanks to franklin tan SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS 16.920J/SMA 5212 Numerical Methods for PDEs Thanks to Franklin Tan Finite Differences: Parabolic Problems B. C. Khoo Lecture 5
Outline Governing Equation Stability analysis 3 Examples Relationship between o and d hh Implicit Time-Marching Scheme Summary SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS Outline • Governing Equation • Stability Analysis • 3 Examples • Relationship between σ and λh • Implicit Time-Marching Scheme • Summary 2
Governing Equation Consider the parabolic pde in 1-D x∈|0,元 at subject to u=uo at x=0, u=u, atx=T If U= viscosity > Diffusion Equation If u= thermal conductivity > Heat Conduction Equation SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS [ 2 2 0, u x t υ π ∂ = ∂ ∂ Governing Equation 3 Consider the Parabolic PDE in 1-D 0 subject to uu at x 0, u u at x = = π = π • If υ ≡ viscosity → Diffusion Equation • If υ ≡ thermal conductivity → Heat Conduction Equation x = 0 x = π u 0 π ux ( , t) = ] u x ∂ ∈ = u ?
Discretization Stability Analysis Keeping time continuous, we carry out a spatial discretization of the rhs of -U at a 0 There is a total of N+l grid points such that x, =jAx j=0,1,2,…,NV SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS Stability Analysis Discretization 4 Keeping time continuous, we carry out a spatial discretization of the RHS of There is a total of 1 grid points such that , 0,1, 2,...., N j j x j + = ∆ = x = 0 x = π 0 x 1 x 3 x N 1 x − Nx 2 2 u t υ ∂ ∂ = ∂ ∂ x N u x
Discretization Stability Analysis Use the central difference scheme for 1-21+1 +O(△x2) which is second-order accurate Schemes of other orders of accuracy may be constructed SMA-HPC 2002 NUS
SMA-HPC ©2002 NUS Stability Analysis Discretization 5 2 1 2 2 2 ( j j j u u u Ox x ∂ + − = ∆ ∂ 2 Us 2 e the Central Difference Scheme for u x ∂ ∂ which is second-order accurate. • Schemes of other orders of accuracy may be constructed. 1 2 ) u j x + − + ∆