16.920J/SMA 5212 Numerical Methods for PDEs OUTLINE Governing equation Stability Analysis 3 Examples Relationship between oand /h Implicit Time-Marching Scheme Summary Side 2 GOVERNING EQUATION Consider the parabolic pde in 1-D du a2u ∈[0.x] subject to u=lo atx=0, u=u atx=Tt Ifu≡ viscosity→ Diffusion Equation If u=thermal conductivity Heat Conduction Equation Side 3 STABILITY ANALYSIS Discretization Keeping time continuous, we carry out a spatial discretization of the rhs of du a at dx 0 X=丌 216.920J/SMA 5212 Numerical Methods for PDEs 2 OUTLINE • Governing Equation • Stability Analysis • 3 Examples • Relationship between σ and λh • 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 u u at x 0, u u at x = = = π = π x = 0 x = π 0 u uπ u ( x,t) = ? 2 2 u u t x υ ∂ ∂ = ∂ ∂ x = 0 x = π 0 x 1 x 2 x N 1 x − N x